new research on graph theory

Suggestions or feedback?

MIT News | Massachusetts Institute of Technology

• Machine learning
• Sustainability
• Black holes
• Classes and programs

Departments

• Aeronautics and Astronautics
• Brain and Cognitive Sciences
• Architecture
• Political Science
• Mechanical Engineering

Centers, Labs, & Programs

• Abdul Latif Jameel Poverty Action Lab (J-PAL)
• Picower Institute for Learning and Memory
• Lincoln Laboratory
• School of Architecture + Planning
• School of Engineering
• School of Humanities, Arts, and Social Sciences
• Sloan School of Management
• School of Science
• MIT Schwarzman College of Computing

MIT engineers’ new theory could improve the design and operation of wind farms

Press contact :, media download.

*Terms of Use:

Images for download on the MIT News office website are made available to non-commercial entities, press and the general public under a Creative Commons Attribution Non-Commercial No Derivatives license . You may not alter the images provided, other than to crop them to size. A credit line must be used when reproducing images; if one is not provided below, credit the images to "MIT."

Previous image Next image

The blades of propellers and wind turbines are designed based on aerodynamics principles that were first described mathematically more than a century ago. But engineers have long realized that these formulas don’t work in every situation. To compensate, they have added ad hoc “correction factors” based on empirical observations.

Now, for the first time, engineers at MIT have developed a comprehensive, physics-based model that accurately represents the airflow around rotors even under extreme conditions, such as when the blades are operating at high forces and speeds, or are angled in certain directions. The model could improve the way rotors themselves are designed, but also the way wind farms are laid out and operated. The new findings are described today in the journal Nature Communications , in an open-access paper by MIT postdoc Jaime Liew, doctoral student Kirby Heck, and Michael Howland, the Esther and Harold E. Edgerton Assistant Professor of Civil and Environmental Engineering.

“We’ve developed a new theory for the aerodynamics of rotors,” Howland says. This theory can be used to determine the forces, flow velocities, and power of a rotor, whether that rotor is extracting energy from the airflow, as in a wind turbine, or applying energy to the flow, as in a ship or airplane propeller. “The theory works in both directions,” he says.

Because the new understanding is a fundamental mathematical model, some of its implications could potentially be applied right away. For example, operators of wind farms must constantly adjust a variety of parameters, including the orientation of each turbine as well as its rotation speed and the angle of its blades, in order to maximize power output while maintaining safety margins. The new model can provide a simple, speedy way of optimizing those factors in real time.

“This is what we’re so excited about, is that it has immediate and direct potential for impact across the value chain of wind power,” Howland says.

Modeling the momentum

Known as momentum theory, the previous model of how rotors interact with their fluid environment — air, water, or otherwise — was initially developed late in the 19th century. With this theory, engineers can start with a given rotor design and configuration, and determine the maximum amount of power that can be derived from that rotor — or, conversely, if it’s a propeller, how much power is needed to generate a given amount of propulsive force.

Momentum theory equations “are the first thing you would read about in a wind energy textbook, and are the first thing that I talk about in my classes when I teach about wind power,” Howland says. From that theory, physicist Albert Betz calculated in 1920 the maximum amount of energy that could theoretically be extracted from wind. Known as the Betz limit, this amount is 59.3 percent of the kinetic energy of the incoming wind.

But just a few years later, others found that the momentum theory broke down “in a pretty dramatic way” at higher forces that correspond to faster blade rotation speeds or different blade angles, Howland says. It fails to predict not only the amount, but even the direction of changes in thrust force at higher rotation speeds or different blade angles: Whereas the theory said the force should start going down above a certain rotation speed or blade angle, experiments show the opposite — that the force continues to increase. “So, it’s not just quantitatively wrong, it’s qualitatively wrong,” Howland says.

The theory also breaks down when there is any misalignment between the rotor and the airflow, which Howland says is “ubiquitous” on wind farms, where turbines are constantly adjusting to changes in wind directions. In fact, in an  earlier paper in 2022, Howland and his team found that deliberately misaligning some turbines slightly relative to the incoming airflow within a wind farm significantly improves the overall power output of the wind farm by reducing wake disturbances to the downstream turbines.

In the past, when designing the profile of rotor blades, the layout of wind turbines in a farm, or the day-to-day operation of wind turbines, engineers have relied on ad hoc adjustments added to the original mathematical formulas, based on some wind tunnel tests and experience with operating wind farms, but with no theoretical underpinnings.

Instead, to arrive at the new model, the team analyzed the interaction of airflow and turbines using detailed computational modeling of the aerodynamics. They found that, for example, the original model had assumed that a drop in air pressure immediately behind the rotor would rapidly return to normal ambient pressure just a short way downstream. But it turns out, Howland says, that as the thrust force keeps increasing, “that assumption is increasingly inaccurate.”

And the inaccuracy occurs very close to the point of the Betz limit that theoretically predicts the maximum performance of a turbine — and therefore is just the desired operating regime for the turbines. “So, we have Betz’s prediction of where we should operate turbines, and within 10 percent of that operational set point that we think maximizes power, the theory completely deteriorates and doesn’t work,” Howland says.

Through their modeling, the researchers also found a way to compensate for the original formula’s reliance on a one-dimensional modeling that assumed the rotor was always precisely aligned with the airflow. To do so, they used fundamental equations that were developed to predict the lift of three-dimensional wings for aerospace applications.

The researchers derived their new model, which they call a unified momentum model, based on theoretical analysis, and then validated it using computational fluid dynamics modeling. In followup work not yet published, they are doing further validation using wind tunnel and field tests.

Fundamental understanding

One interesting outcome of the new formula is that it changes the calculation of the Betz limit, showing that it’s possible to extract a bit more power than the original formula predicted. Although it’s not a significant change — on the order of a few percent — “it’s interesting that now we have a new theory, and the Betz limit that’s been the rule of thumb for a hundred years is actually modified because of the new theory,” Howland says. “And that’s immediately useful.” The new model shows how to maximize power from turbines that are misaligned with the airflow, which the Betz limit cannot account for.

The aspects related to controlling both individual turbines and arrays of turbines can be implemented without requiring any modifications to existing hardware in place within wind farms. In fact, this has already happened, based on earlier work from Howland and his collaborators two years ago that dealt with the wake interactions between turbines in a wind farm, and was based on the existing, empirically based formulas.

“This breakthrough is a natural extension of our previous work on optimizing utility-scale wind farms,” he says, because in doing that analysis, they saw the shortcomings of the existing methods for analyzing the forces at work and predicting power produced by wind turbines. “Existing modeling using empiricism just wasn’t getting the job done,” he says.

In a wind farm, individual turbines will sap some of the energy available to neighboring turbines, because of wake effects. Accurate wake modeling is important both for designing the layout of turbines in a wind farm, and also for the operation of that farm, determining moment to moment how to set the angles and speeds of each turbine in the array.

Until now, Howland says, even the operators of wind farms, the manufacturers, and the designers of the turbine blades had no way to predict how much the power output of a turbine would be affected by a given change such as its angle to the wind without using empirical corrections. “That’s because there was no theory for it. So, that’s what we worked on here. Our theory can directly tell you, without any empirical corrections, for the first time, how you should actually operate a wind turbine to maximize its power,” he says.

Because the fluid flow regimes are similar, the model also applies to propellers, whether for aircraft or ships, and also for hydrokinetic turbines such as tidal or river turbines. Although they didn’t focus on that aspect in this research, “it’s in the theoretical modeling naturally,” he says.

The new theory exists in the form of a set of mathematical formulas that a user could incorporate in their own software, or as an open-source software package that can be freely downloaded from GitHub . “It’s an engineering model developed for fast-running tools for rapid prototyping and control and optimization,” Howland says. “The goal of our modeling is to position the field of wind energy research to move more aggressively in the development of the wind capacity and reliability necessary to respond to climate change.”

The work was supported by the National Science Foundation and Siemens Gamesa Renewable Energy.

Share this news article on:

Related links.

• Unified momentum model software
• Howland Lab
• Department of Civil and Environmental Engineering

Related Topics

• Civil and environmental engineering
• Renewable energy
• Climate change

Related Articles

Michael Howland gives wind energy a lift

A new method boosts wind farms’ energy output, without new equipment

The answer may be blowing in the wind

Previous item Next item

More MIT News

A framework for solving parabolic partial differential equations

Read full story →

Designing better delivery for medical therapies

Making a measurable economic impact

Faces of MIT: Jessica Tam

First AI + Education Summit is an international push for “AI fluency”

President Kornbluth welcomes the Class of 2028

• More news on MIT News homepage →

Massachusetts Institute of Technology 77 Massachusetts Avenue, Cambridge, MA, USA

• Map (opens in new window)
• Events (opens in new window)
• People (opens in new window)
• Careers (opens in new window)
• Accessibility
• Social Media Hub
• MIT on Facebook
• MIT on YouTube
• MIT on Instagram

Research on the Construction Method of Industry Information Knowledge Graph for Electricity Customers

New citation alert added.

This alert has been successfully added and will be sent to:

You will be notified whenever a record that you have chosen has been cited.

To manage your alert preferences, click on the button below.

New Citation Alert!

Please log in to your account

Information & Contributors

Bibliometrics & citations, index terms.

Applied computing

Enterprise computing

Enterprise modeling

Recommendations

Removable edges in a 5-connected graph and a construction method of 5-connected graphs.

An edge e of a k-connected graph G is said to be a removable edge if G@?e is still k-connected. A k-connected graph G is said to be a quasi (k+1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-...

Removable edges in a k-connected graph and a construction method for k-connected graphs

An edge e of a k-connected graph G is said to be a removable edge if G@?e is still k-connected, where G@?e denotes the graph obtained from G by deleting e to get G-e and, for any end vertex of e with degree k-1 in G-e, say x, deleting x and then adding ...

Research on Knowledge Graph Construction for Python Programming Language

As the core and foundation of artificial intelligence, knowledge graph provides new power to smart education. With the popularization and low age of learning programming languages, building knowledge graphs for them is especially crucial for ...

Information

Published in.

Association for Computing Machinery

New York, NY, United States

Publication History

Permissions, check for updates, author tags.

• Electricity Customers
• Knowledge Graph
• User classification
• Research-article
• Refereed limited

Contributors

Other metrics, bibliometrics, article metrics.

• 0 Total Citations
• 0 Total Downloads
• Downloads (Last 12 months) 0
• Downloads (Last 6 weeks) 0

View Options

Login options.

Check if you have access through your login credentials or your institution to get full access on this article.

Full Access

View options.

View or Download as a PDF file.

View online with eReader .

HTML Format

View this article in HTML Format.

Share this Publication link

Copying failed.

Share on social media

Affiliations, export citations.

• Please download or close your previous search result export first before starting a new bulk export. Preview is not available. By clicking download, a status dialog will open to start the export process. The process may take a few minutes but once it finishes a file will be downloadable from your browser. You may continue to browse the DL while the export process is in progress. Download
• Download citation
• Copy citation

We are preparing your search results for download ...

We will inform you here when the file is ready.

Your file of search results citations is now ready.

Your search export query has expired. Please try again.

Anthropology

Meet the 2024-25 first-year graduate cohort.

The Department of Anthropology is thrilled to welcome its 2024-25 first-year graduate cohort to Brown! From family ties and kinship to Black Feminist Theory to global supply chains — this year's cohort brings a full spectrum of research interests to College Hill.

Pictured from left to right (top row): Avani Ashtekar, Juben Rabbani, Eliza Ge, Macie Clerkley Pictured from left to right (bottom row): Paul Tamburro, Emma Kahn, Grace Horseman, Yuan-Jhen (Emerson) Lee, Stella Mojisola Oyekanmi

Click through the following profiles to learn more about each new community member:

Avani Ashtekar

Macie Clerkley

Grace Horseman

Stella Mojisola Oyekanmi

Juben Rabbani

Emerson Lee

Paul Tamburro

How humpback whales use bubbles as a tool

A new study confirms a theory that humpbacks are a tool-using species, as they deploy nets of bubbles to catch fish and krill.

Chimpanzees use sticks to fish for termites , sea otters crack open clams with a rock and dolphins use sea sponges on their noses for protection while foraging on the ocean floor. A new study adds humpback whales to the list of non-human species that use tools. Humpbacks may not only use a tool, but create it from their environment by blowing bubbles.

Around the world, humpback whales ( Megaptera novaeangliae ) use bubble-nets to trap certain prey such as krill, herring, and young salmon, sometimes in coordinated groups and at times alone. The whales dive down below their prey and swim in circles while releasing bubbles from their blowholes to create a rising curtain. The curtain creates a visual barrier that tricks the prey into thinking there’s no escape. Once the prey is tightly corralled, the whales lunge through the bubble-net with open mouths to swallow their meal. This feeding behavior has been observed for decades , but the precise mechanics behind it are difficult to study and have long remained a mystery.

While watching humpbacks feeding “it looks like a big scattering of bubbles and it doesn't look like it's very structured,” says Andrew Szabo, a marine ecologist who leads the Alaska Whale Foundation and an author on the study. But that all changes when you add drones and underwater cameras, he says.

While tool-use in animals can sometimes be challenging to define, scientists now tend to think about describing tool use as using an external object that isn’t attached to anything to change the shape, position, or condition of something else. It’s been suggested before that bubble-netting is using a tool, but “this paper strengthens that position,” says Janet Mann , a marine mammal biologist at Georgetown University in Washington D.C. who has studied aquatic tool-use extensively.

Feeding frenzy

To get a better look at the behavior, Szabo and the team used 20-foot poles to attach specialized underwater suction tags on whales in northern Southeast Alaskan waters. These tags, equipped with 4K video cameras, hydrophones and sensors to record movement in three dimensions, as well as temperature and depth, collected data for up to 24 hours before detaching. The scientists combined the tag data with aerial footage captured by drones to precisely measure the timing, structure, and size of the bubble nets made by solitary whales.

And it turns out that these gentle giants adjust the speed and spacing of their bubble emissions to trap prey more effectively, the researchers report August 21 in Proceedings of the Royal Society Open Science . By altering the bubble rings, the whales may be able to catch seven times the prey on average in just one gulp. The whales are conserving energy, having to lunge fewer times, says Szabo, who is also a National Geographic Explorer . This efficiency is crucial for humpbacks, as they migrate thousands of miles and need to capture enough food during the summer and fall in Alaska to sustain themselves throughout the year.

“They have so much control over how they're doing this,” says Lars Bejder, another author a the study and marine mammal biologist who directs the University of Hawai’i at Manoa’s Marine Mammal Research Program . “They are increasing the frequency of the pulses as the net gets smaller, to lessen the mesh size of how the prey can escape. And that's really cool.”

For the team, humpbacks’ ability to use bubble-nets as tools speaks to the whales’ cognition and complexity, which is often overshadowed by other marine mammals, like dolphins, Szabo says. “They are remarkable animals, doing remarkable things.”

You May Also Like

Why did a humpback whale capsize a boat? Here's what really happened.

What is ‘kelping’? Why whales are making hats out of seaweed

First-ever photos show humpback whales mating—and they’re males

Caloric capture.

Interestingly, the whales did not consistently use bubble-nets. In Alaska, only about five to 10 percent of the whales bubble-net feed. “It’s certainly the rarity, it's not the commonality,” Bejder says. “This is true in all populations, as well.”

During the three-day period when the team deployed tags, they observed 70 to 80 whales engaging in bubble-net feeding. However, just a week later, the same whales had stopped using this tactic. Why was this the case?

When and where humpback whales use bubbles may have to do with prey density. “It takes a long time to deploy these nets, and if the food is sufficiently dense where you don't need to use a net, you might actually do better not using it,” Szabo says. However, if the prey aren’t very dense, using bubble-nets allows the whales to exploit an otherwise unavailable resource. “You can actually use these nets to make something that was not profitable, profitable to feed on,” he says.

While impressive, it’s not particularly surprising, says Jan Straley, a biologist and professor emerita at the University of Alaska Southeast who has studied humpback whales since 1979. Humpback whales are known to vary where they deploy bubble nets based on prey location, environmental conditions and density. She has observed young whales learning from their moms and appear to pick up new techniques from peers.

“I think these whales are really, really good at knowing their environment and knowing the physics of their environment. They know the physical properties of the water column and how sound travels,” she says. “They're really smart for their world.”

For instance, in a region outside Glacier Bay National Park in Alaska, humpback whales feed in coordinated groups but do not use bubble-nets. Instead, they utilize tides and currents to corral fish.

In the 1980s, New England’s humpback whales developed a technique called lobtail feeding , where they slap their tails before bubble feeding. This behavior may have started as the whales changed their diet from herring to sand lance and spread through social learning.

The humpbacks' ability to change their feeding strategies and use tools to access otherwise unreachable prey might explain why they have fared better than other large whales since the whaling era. This adaptability could also give them a better chance of adjusting to climate change—   as long as the prey don't disappear entirely. “Prey is declining. We know this,” says Szabo. “Whales are getting skinnier.”

Related Topics

• HUMPBACK WHALE
• MARINE MAMMALS
• MARINE BIOLOGY

Whales can sing underwater without drowning—now we know how

20 of the coolest travel adventures for 2024

How billions of dollars and cutting-edge tech are revolutionizing ocean exploration

How scientists are piecing together a sperm whale ‘alphabet’

A humpback whale swam halfway around the world. His name is Frodo.

• Best of the World
• Environment
• Paid Content

History & Culture

• History & Culture
• Terms of Use
• Privacy Policy
• Your US State Privacy Rights
• Children's Online Privacy Policy
• Interest-Based Ads
• About Nielsen Measurement
• Do Not Sell or Share My Personal Information
• Nat Geo Home
• Attend a Live Event
• Book a Trip
• Inspire Your Kids
• Shop Nat Geo
• Visit the D.C. Museum
• Learn About Our Impact
• Support Our Mission
• Advertise With Us
• Customer Service
• Renew Subscription
• Manage Your Subscription
• Work at Nat Geo
• Sign Up for Our Newsletters
• Contribute to Protect the Planet

Copyright © 1996-2015 National Geographic Society Copyright © 2015-2024 National Geographic Partners, LLC. All rights reserved

Numbers, Facts and Trends Shaping Your World

Read our research on:

Full Topic List

Regions & Countries

• Publications
• Our Methods
• Short Reads
• Tools & Resources

Read Our Research On:

Harris Energizes Democrats in Transformed Presidential Race

1. the presidential matchup: harris, trump, kennedy, table of contents.

• Other findings: Both Harris and Trump are viewed more favorably than a few months ago
• Voting preferences among demographic groups
• How have voters shifted their preferences since July?
• Harris’ supporters back her more strongly than Biden’s did last month
• Large gap in motivation to vote emerges between the candidates’ younger supporters
• Harris and Trump have gained ground with their own coalitions
• Share of ‘double negatives’ drops significantly with change in presidential candidates
• Views of Biden have changed little since his withdrawal from the 2024 presidential race
• Acknowledgments
• The American Trends Panel survey methodology

Nationally, Vice President Kamala Harris and former President Donald Trump are essentially tied among registered voters in the current snapshot of the presidential race: 46% prefer Harris, 45% prefer Trump and 7% prefer Robert F. Kennedy Jr.

Following Biden’s exit from the race, Trump’s support among voters has remained largely steady (44% backed him in July against Biden, while 45% back him against Harris today). However, Harris’ support is 6 percentage points higher than Biden’s was in July . In addition to holding on to the support of those who backed Biden in July, Harris’ bump has largely come from those who had previously said they supported or leaned toward Kennedy.

Harris performs best among the same demographic groups as Biden. But this coalition of voters is now much more likely to say they strongly support her: In July, 43% of Biden’s supporters characterized their support as strong – today, 62% of Harris’ do.

Overall, many of the same voting patterns that were evident in the Biden-Trump matchup from July continue to be seen today. Harris fares better than Trump among younger voters, Black voters, Asian voters and voters with college degrees. By comparison, the former president does better among older voters, White voters and voters without a college degree.

But Harris performs better than Biden across many of these groups – making the race tighter than it was just a few weeks ago.

• In July, women’s presidential preferences were split: 40% backed Biden, 40% preferred Trump and 17% favored Kennedy. With Harris at the top of the ticket, 49% of women voters now support her, while 42% favor Trump and 7% back Kennedy.
• Among men, Trump draws a similar level of support as he did in the race against Biden (49% today, compared with 48% in July). But the share of men who now say they support Harris has grown (to 44% today, up from 38% last month). As a result, Trump’s 10-point lead among men has narrowed to a 5-point lead today.

Race and ethnicity

Harris has gained substantial ground over Biden’s position in July among Black, Hispanic and Asian voters. Most of this movement is attributable to declining shares of support for Kennedy. Trump performs similarly among these groups as he did in July.

• 77% of Black voters support or lean toward Harris. This compares with 64% of Black voters who said they backed Biden a few weeks ago. Trump’s support is unchanged (13% then vs. 13% today). And while 21% of Black voters supported Kennedy in July, this has dropped to 7% in the latest survey.
• Hispanic voters now favor Harris over Trump by a 17-point margin (52% to 35%). In July, Biden and Trump were tied among Hispanic voters with 36% each.
• By about two-to-one, Asian voters support Harris (62%) over Trump (28%). Trump’s support among this group is essentially unchanged since July, but the share of Asian voters backing Harris is 15 points higher than the share who backed Biden in July.
• On balance, White voters continue to back Trump (52% Trump, 41% Harris), though that margin is somewhat narrower than it was in the July matchup against Biden (50% Trump, 36% Biden).

While the age patterns present in the Harris-Trump matchup remain broadly the same as those in the Biden-Trump matchup in July, Harris performs better across age groups than Biden did last month. That improvement is somewhat more pronounced among voters under 50 than among older voters.

• Today, 57% of voters under 30 say they support Harris, while 29% support Trump and 12% prefer Kennedy. In July, 48% of these voters said they backed Biden. Trump’s support among this group is essentially unchanged. And 12% now back Kennedy, down from 22% in July.
• Voters ages 30 to 49 are now about evenly split (45% Harris, 43% Trump). This is a shift from a narrow Trump lead among this group in July.
• Voters ages 50 and older continue to tilt toward Trump (50% Trump vs. 44% Harris).

With Harris now at the top of the Democratic ticket, the race has become tighter.

Much of this is the result of shifting preferences among registered voters who, in July, said they favored Kennedy over Trump or Biden.

Among the same group of voters surveyed in July and early August, 97% of those who backed Biden a few weeks ago say they support or lean toward Harris today. Similarly, Trump holds on to 95% of those who supported him a few weeks ago.

But there has been far more movement among voters who previously expressed support for Kennedy. While Kennedy holds on to 39% of those who backed him in July, the majority of these supporters now prefer one of the two major party candidates: By about two-to-one, those voters are more likely to have moved to Harris (39%) than Trump (20%). This pattern is evident across most voting subgroups.

In July, Trump’s voters were far more likely than Biden’s voters to characterize their support for their candidate as “strong” (63% vs. 43%). But that gap is no longer present in the Harris-Trump matchup.

Today, 62% of Harris voters say they strongly support her, while about a third (32%) say they moderately support her. Trump’s voters are just about as likely to say they strongly back him today as they were in July (64% today, 63% then).

Kennedy’s voters make up a smaller share of voters today than a month ago – and just 18% of his voters say they strongly support him, similar to the 15% who said the same in July.

Across demographic groups, strong support for Harris is higher than it was for Biden

Among women voters who supported Biden in July, 45% said they did so strongly. That has grown to 65% today among women voters who support Harris.

Increased intensity of support is similar among men voters who back the Democratic candidate: In July, 42% of men voters who supported Biden said they did so strongly. This has since grown to 59% of Harris’ voters who are men.

Across racial and ethnic groups, Harris’ supporters are more likely than Biden’s were to say they back their candidates strongly.

Among White voters, 43% who supported Biden in July did so strongly. Today, Harris’ strong support among White voters sits at 64%.

A near identical share of Harris’ Black supporters (65%) characterize their support for her as strong today. This is up from the 52% of Biden’s Black supporters who strongly backed him in July. Among Harris’ Hispanic supporters, 56% support her strongly, while 45% of Asian Harris voters feel the same. Strong support for Harris among these voters is also higher than it was for Biden in July.

Across all age groups, Harris’ strength of support is higher than Biden’s was. But the shift from Biden is less pronounced among older Democratic supporters than among younger groups.

Still, older Harris voters are more likely than younger Harris voters to describe their support as strong. For instance, 51% of Harris’ voters under 50 say they strongly support her, while 71% of Harris supporters ages 50 and older characterize their support as strong.

Today, about seven-in-ten of both Trump supporters (72%) and Harris supporters (70%) say they are extremely motivated to vote.

Motivation to vote is higher in both the Democratic and Republican coalitions than it was in July .

These shifts have occurred across groups but are more pronounced among younger voters.

Today, half of voters under 30 say they are extremely motivated to vote, up 16 points since July. Motivation is up 11 points among voters ages 30 to 49 and 50 to 64, and up 6 points among those ages 65 and older.

Among the youngest voters, the increased motivation to vote is nearly all driven by shifts among Democratic supporters.

• In July, 38% of 18- to 29-year-old Trump voters said they were extremely motivated to vote. Today, a similar share of his voters (42%) report that level of motivation.
• But 18- to 29-year-old Harris supporters are far more likely to say they are extremely motivated to vote than Biden’s supporters in this age group were about a month ago. Today, 61% of Harris’ voters under 30 say this. In July, 42% of voters under 30 who supported Biden said they were extremely motivated to vote.

Sign up for our weekly newsletter

Fresh data delivery Saturday mornings

Sign up for The Briefing

Weekly updates on the world of news & information

• Donald Trump
• Election 2024
• Kamala Harris
• More Leaders
• Political & Civic Engagement

The Political Values of Harris and Trump Supporters

As robert f. kennedy jr. exits, a look at who supported him in the 2024 presidential race, many americans are confident the 2024 election will be conducted fairly, but wide partisan differences remain, joe biden, public opinion and his withdrawal from the 2024 race, amid doubts about biden’s mental sharpness, trump leads presidential race, most popular, report materials.

• August 2024 Presidential Preference Detailed Tables
• Questionnaire

901 E St. NW, Suite 300 Washington, DC 20004 USA (+1) 202-419-4300 | Main (+1) 202-857-8562 | Fax (+1) 202-419-4372 |  Media Inquiries

Research Topics

• Email Newsletters

ABOUT PEW RESEARCH CENTER  Pew Research Center is a nonpartisan fact tank that informs the public about the issues, attitudes and trends shaping the world. It conducts public opinion polling, demographic research, media content analysis and other empirical social science research. Pew Research Center does not take policy positions. It is a subsidiary of  The Pew Charitable Trusts .

© 2024 Pew Research Center

• Share full article

Advertisement

Supported by

Scientists Have a New Theory About Why Orcas Are Attacking Boats

A pod of orcas damaged a boat and left its two-person crew stranded. It was the latest in a string of attacks that research suggests could be used for hunting practice.

By Lynsey Chutel

Reporting from London

The orcas have struck again — this time ramming a sailboat off Spain’s northwest coast, rescue workers said on Tuesday.

A pod of orcas damaged the rudder of a sailboat, leaving its two-person crew stranded in the waters off Cape Finisterre Sunday, according to an emailed statement from the rescue workers. It is the latest in a string of attacks by pods of orcas swimming around the Iberian Peninsula.

While the sailboat, the Amidala, did not sink, pods of orcas have sunk several vessels in recent years. Researchers still do not know whether the attacks are playful or malicious, but a new theory based on studying the troublesome pods of orcas suggests that they could be using the boats as practice targets for new hunting techniques. Other competing theories still exist.

Regardless of the orcas’ intentions, the behavior is enough to worry sailors journeying in the highly trafficked waters around North Africa, Spain and Portugal.

The Amidala, manned by a crew of two Belgians, encountered an unknown number of orcas on Sunday afternoon. They sent a mayday distress call to the Finisterre Maritime Rescue Center, which towed the vessel back to shore, the center said.

The sailboat’s damaged rudder, and poor weather conditions in the area, made the rescue more arduous, with waves reaching up to nearly 10 feet and winds hitting speeds of 40 miles per hour. A female crew member on the Amidala suffered injuries to her hand as the sailboat was being towed, and she was transferred to a rescue vessel, the rescue center said. After more than four hours, the Amidala made it back to shore.

We are having trouble retrieving the article content.

Please enable JavaScript in your browser settings.

Thank you for your patience while we verify access. If you are in Reader mode please exit and  log into  your Times account, or  subscribe  for all of The Times.

Thank you for your patience while we verify access.

Already a subscriber?  Log in .

Want all of The Times?  Subscribe .

Who will win the 2024 election? What the polls say ahead of DNC

Since taking the reins last month as presumptive nominee of the Democratic party, Vice President Kamala Harris has seen a sharp rise in the polls over Biden’s performance.

The endgame of the presidential race is still months away and it’s too early to draw hard conclusions from current polling numbers — which are likely to shift and sway over the next three months.

But several recent nationwide polls show Harris with a slight lead over Republican nominee former President Donald Trump, even as the average among the critical battleground states remains tight.

Will Trump or Harris win?

A CBS News and YouGov poll of likely voters concluded on August 16 shows Kamala Harris polling at 49% nationally to Donald Trump’s 47%.

CBS’s polling also shows that the percentage of registered Democrats who say that they’ll vote in November is up 6% from a month ago, from 81% to 87%, while that same measure is down for Republicans, from 90% to 88%.

Among likely voters, CBS found that top issues influencing votes for president were the economy (83%), inflation (76%), the state of democracy (74%), crime (62%), gun policy (58%), the U.S.-Mexico border (56%) and abortion (51%).

There’s a relatively large gap between the two presidential candidates in terms of how familiar voters are with their platforms in the CBS poll. A full 86% of respondents said they knew what Trump stands for but only 64% said the same of Harris.

This could perhaps give Harris an opportunity to expand her appeal as she looks to further define her political persona and her policy platform under the spotlight of this week’s Democratic National Convention.

Trump is still in the lead among respondents who say that their main voting issues are the economy, inflation and border issues.

Republican vice-presidential nominee JD Vance cast spurious doubt on the reliability of polling broadly, telling Fox News that “The media uses fake polls to drive down Republican turnout and to create dissension and conflict with Republican voters.”

The Trump campaign has, however, plugged polling numbers before, when they were in his favor.

Emerson College shows Harris up, at 48% compared to Trump’s 44%.

An August 13 poll by The Economist has Harris at 46% and Trump at 44%.

A Washington Post /ABC News/Ipsos poll, also concluded August 13 shows Harris in the lead, 47% to 44%.

The New York Times ’ national polling average puts Harris ahead with 46% against Trump’s 45%.

A smaller number of polls still show Trump ahead—a poll from RMG Research and the Napolitan Institute shows Trump leading, 49% to 47% and another from J.L. Partners and Dailymail.com show him up 43% to 41%.

In a poll conducted August 9-12, Fox News had both Trump and Harris at 45%.

Who is winning on the state level?

The picture becomes more complicated when you start looking at individual states. Because of the electoral college, being ahead nationally doesn’t necessarily translate into taking office in January.

According to New York Times averages, Harris had pulled ahead of Trump in several important Rust Belt battleground states including Michigan (49-47%), Wisconsin (49-47%) and Pennsylvania (49-48%).

The Times average shows Trump holding his lead in certain Sun Belt states including Georgia, where he’s ahead 49% to 46% and in Arizona Trump and Harris both have 47%.

However, a Times/Siena poll from Thursday has Harris winning in Arizona, 47% to 43%, though it still shows Trump up in Nevada (46 to 44%) and in Georgia (47 to 44%.)

Emerson College shows Trump up very slightly in Pennsylvania, with both him and Harris rounded to 47%. Polls earlier this month from Franklin and Marshall College and Quinnipiac University, however, both show Harris up in Pennsylvania by 3 points, 46% to 43% and 48% to 45%, respectively.

RECENT ADVANCES IN GRAPH THEORY AND ITS APPLICATIONS

• 10(3):1407-1412

• B.M.S Institute of Technology & Mnagement
• This person is not on ResearchGate, or hasn't claimed this research yet.

• GITAM School of Technology,GITAM University Bengalur

• APS College Engineering , Bengaluru-560082

Discover the world's research

• 25+ million members
• 160+ million publication pages
• 2.3+ billion citations

• Arudra Annepu

• Pallavi Singh

• Ana Carolina Coelho Lopes
• Tiago Alcântara Ribeiro
• António Fernando Macedo Ribeiro
• Farah Maulidya Fatimah

• Nandini S R

• Anand Sharma

• S Pruthvi Sagar
• K Khalandar

• Salwan A. Abdlwahab

• Vishwas V. Rudraswamimath

• Alessandra Serrau

• Ramin Zabih

• Wei-Chieh Ke
• Chin-Hsien Tsai

• Recruit researchers
• Join for free
• Login Email Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google Welcome back! Please log in. Email · Hint Tip: Most researchers use their institutional email address as their ResearchGate login Password Forgot password? Keep me logged in Log in or Continue with Google No account? Sign up

Research Trends in Graph Theory and Applications

• © 2021
• Daniela Ferrero   ORCID: https://orcid.org/0000-0003-4675-1814 0 ,
• Leslie Hogben   ORCID: https://orcid.org/0000-0003-1673-3789 1 ,
• Sandra R. Kingan   ORCID: https://orcid.org/0000-0002-8232-5540 2 ,
• Gretchen L. Matthews   ORCID: https://orcid.org/0000-0002-8977-8171 3

Department of Mathematics, Texas State University, San Marcos, USA

You can also search for this editor in PubMed   Google Scholar

Department of Mathematics, Iowa State University, Ames, USA

Department of mathematics, brooklyn college and the graduate center, city university of new york, new york, usa, department of mathematics, virginia polytechnic institute and state university, blacksburg, usa.

• Features research in a broad variety of problems in different areas of graph theory
• Each chapter offers an introduction to a graph theory topic of current research, aiming at clarity and high-quality exposition while emphasizing recent advances and open problems
• Presents concepts and ideas thoroughly and with details
• Each chapter corresponds to research proposed and led by prominent research experts in each field

Part of the book series: Association for Women in Mathematics Series (AWMS, volume 25)

2989 Accesses

4 Citations

2 Altmetric

This is a preview of subscription content, log in via an institution to check access.

Access this book

Subscribe and save.

• Get 10 units per month
• Download Article/Chapter or eBook
• 1 Unit = 1 Article or 1 Chapter
• Cancel anytime
• Available as EPUB and PDF
• Read on any device
• Instant download
• Own it forever
• Compact, lightweight edition
• Dispatched in 3 to 5 business days
• Free shipping worldwide - see info
• Durable hardcover edition

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

About this book

Similar content being viewed by others.

Graph Theory – A Survey on the Occasion of the Abel Prize for László Lovász

Introduction

The ubiquity of large graphs and surprising challenges of graph processing: extended survey

• graph symmetries
• metric dimension
• graph searching
• data storage
• power domination

Table of contents (6 chapters)

Front matter, finding long cycles in balanced tripartite graphs: a first step.

• Gabriela Araujo-Pardo, Zhanar Berikkyzy, Jill Faudree, Kirsten Hogenson, Rachel Kirsch, Linda Lesniak et al.

Product Thottling

• Sarah E. Anderson, Karen L. Collins, Daniela Ferrero, Leslie Hogben, Carolyn Mayer, Ann N. Trenk et al.

Analysis of Termatiko Sets in Measurement Matrices

• Katherine F. Benson, Jessalyn Bolkema, Kathryn Haymaker, Christine Kelley, Sandra R. Kingan, Gretchen L. Matthews et al.

The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey

• Nadia Benakli, Novi H. Bong, Shonda Dueck (Gosselin), Beth Novick, Ortrud R. Oellermann

Symmetry Parameters for Mycielskian Graphs

• Debra Boutin, Sally Cockburn, Lauren Keough, Sarah Loeb, K. E. Perry, Puck Rombach

Reconfiguration Graphs for Dominating Sets

• Kira Adaricheva, Chassidy Bozeman, Nancy E. Clarke, Ruth Haas, Margaret-Ellen Messinger, Karen Seyffarth et al.

Editors and Affiliations

Daniela Ferrero

Leslie Hogben

Sandra R. Kingan

Gretchen L. Matthews

About the editors

Bibliographic information.

Book Title : Research Trends in Graph Theory and Applications

Editors : Daniela Ferrero, Leslie Hogben, Sandra R. Kingan, Gretchen L. Matthews

Series Title : Association for Women in Mathematics Series

DOI : https://doi.org/10.1007/978-3-030-77983-2

Publisher : Springer Cham

eBook Packages : Mathematics and Statistics , Mathematics and Statistics (R0)

Copyright Information : The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

Hardcover ISBN : 978-3-030-77982-5 Published: 07 September 2021

Softcover ISBN : 978-3-030-77985-6 Published: 08 September 2022

eBook ISBN : 978-3-030-77983-2 Published: 06 September 2021

Series ISSN : 2364-5733

Series E-ISSN : 2364-5741

Edition Number : 1

Number of Pages : XVIII, 135

Number of Illustrations : 47 b/w illustrations

Topics : Graph Theory , Computer Communication Networks

• Publish with us

Policies and ethics

• Find a journal
• Track your research

• Search this journal
• Search all journals
• View access options
• View profile
• Create profile

Add email alerts

You are adding the following journal to your email alerts

Application of graph theory in the library domain—Building a faceted framework based on a literature review

Introduction, related work, graphs and networks in the library domain, a framework for the application of graph theory in the library domain, framework application, declaration of conflicting interests, biographies, cite article, share options, information, rights and permissions, metrics and citations, figures and tables, citation networks, collaboration networks, content networks, library networks, metadata record networks, existing frameworks and classification schemes, framework principles, framework categories and facets, category 1: node objects, category 1.1: people, category 1.2: documents, category 1.3: journals, category 1.4: words, category 1.5: institutions, category 1.6: countries, category 2: edge definitions, category 2.1: citation, category 2.2: collaboration, category 2.3: similarity, category 2.4: (physical) connection, category 3: research objectives, category 3.14: analyzing information structures, category 3.15: library operations, category 4: graph characteristics, category 4.1: small-world, category 4.2: transitivity, category 4.3: degree distributions, category 5: methodology, category 5.1: network metrics, category 5.2: software tools, category 5.3: algorithms, combining categories.

Defining concepts and finding research desiderata

For simplicity, a set of authors together with a set of co-author pairs will be referred to as a co-author graph , and any graph whose lines represent channels of communication, through which information can be transmitted or exchanged, will be referred to as a communication-graph.

Application domain

Bibliographic data, data sources, possible framework refinements, formal definitions, integrating more research fields, designing standard workflows, download to reference manager.

If you have citation software installed, you can download article citation data to the citation manager of your choice

Share this article

Share with email, share on social media, share access to this article.

Sharing links are not relevant where the article is open access and not available if you do not have a subscription.

For more information view the Sage Journals article sharing page.

Information

Published in.

• Graph theory
• library science
• network studies
• bibliographic data science
• metadata records

Rights and permissions

Affiliations, journals metrics.

This article was published in Journal of Librarianship and Information Science .

Article usage *

Total views and downloads: 2713

* Article usage tracking started in December 2016

Articles citing this one

Receive email alerts when this article is cited

Web of Science: 3 view articles Opens in new tab

Crossref: 0

• Development of intelligent system of global bibliographic search Go to citation Crossref Google Scholar

Figures & Media

View options, view options, access options.

If you have access to journal content via a personal subscription, university, library, employer or society, select from the options below:

I am signed in as:

I can access personal subscriptions, purchases, paired institutional access and free tools such as favourite journals, email alerts and saved searches.

Login failed. Please check you entered the correct user name and password.

Access personal subscriptions, purchases, paired institutional or society access and free tools such as email alerts and saved searches.

loading institutional access options

Click the button below for the full-text content

Alternatively, view purchase options below:

Purchase 24 hour online access to view and download content.

Access journal content via a DeepDyve subscription or find out more about this option.

Also from Sage

• CQ Library Elevating debate opens in new tab
• Sage Data Uncovering insight opens in new tab
• Sage Business Cases Shaping futures opens in new tab
• Sage Campus Unleashing potential opens in new tab
• Sage Knowledge Multimedia learning resources opens in new tab
• Sage Research Methods Supercharging research opens in new tab
• Sage Video Streaming knowledge opens in new tab
• Technology from Sage Library digital services opens in new tab

An editorially independent publication supported by the Simons Foundation.

Get the latest news delivered to your inbox.

Type search term(s) and press enter

• Comment Comments
• Save Article Read Later Read Later

A Very Big Small Leap Forward in Graph Theory

May 2, 2023

DVDP for Quanta Magazine

Introduction

On March 15, intriguing seminar announcements sent rumblings through the field of combinatorics, the mathematical study of counting. Three collaborators planned to give coordinated talks the following day. Julian Sahasrabudhe would address mathematicians in Cambridge, England, while Simon Griffiths would share the news in Rio de Janeiro and Marcelo Campos in São Paulo. All three talks sported identical titles and cryptic, two-sentence abstracts referencing “recent progress on an old problem of Erdős.” While Paul Erdős, a Hungarian mathematician who died in 1996, posed hundreds of problems during his career, combinatorists quickly divined the specific issue that the trio was planning to talk about. Rumors swirled about something called the Ramsey number, one of the most difficult quantities to calculate in all of mathematics.

Ramsey numbers have spawned a whole discipline called Ramsey theory, which looks for inescapable patterns in a huge range of systems.

For example, say you try to spread all the integers among a number of buckets, and you want to avoid placing sequences of evenly spaced numbers in any of the buckets. Ramsey theory shows that you will fail (unless you have infinitely many buckets). The theory can be applied to most anything you can count. Its central lesson is that “you cannot create a completely chaotic system,” said Benny Sudakov, a mathematician at the Swiss Federal Institute of Technology Zurich.

The Ramsey number quantifies how big a paradigmatic example has to be before particular patterns inevitably arise. But despite its centrality, no one has been able to calculate the Ramsey number for all but the simplest instances . The best that they have been able to do is find limits (or bounds) on what it might be. Even then, the upper bound first established by Erdős and a collaborator nearly a century ago had barely budged.

Then, in the March 16 seminars, and in a paper posted later that evening, the researchers announced that they had improved the upper bound on the Ramsey number by an exponential amount.

“We were stuck for a very long time,” said Marcelo Campos.

Gabrielle Pereira

“I was floored,” said Yuval Wigderson , a mathematician at Tel Aviv University, on hearing about the new result. “I was literally shaking for half an hour to an hour.”

The Party Lines

Ramsey theory most commonly asks questions either about the integers or about graphs. A graph, in this context, refers to collections of points called nodes, connected by lines called edges, which can have properties like length or — as in the case of the Ramsey numbers — color.

A complete graph is both complicated and simple — every node is connected to every other node. The Ramsey number describes how many nodes a complete graph must contain to be forced to have a particular structure. Say the edges of a complete graph are assigned one of two colors: red or blue. And say you try to color the edges in a way that avoids connecting a group of nodes with edges of the same color. In 1930, Frank Ramsey proved that if a graph is big enough, it becomes impossible to avoid creating what mathematicians call a monochromatic clique — a group of nodes whose common edges are either all red, or all blue.

How big, exactly, must a graph be before a monochromatic clique is forced to emerge? The answer depends on the size of the clique. Ramsey showed that there exists a number, now called the Ramsey number, representing the minimum number of nodes for which a monochromatic clique of a given size must exist, no matter how the edges are colored.

But the size of the Ramsey number is hard to pin down. In 1935, five years after Ramsey showed it exists, Erdős and George Szekeres provided a new, tighter upper bound on how big the Ramsey number is for a clique of a given size. But since then, mathematicians have barely been able to improve on Erdős and Szekeres’ calculation.

To get a better intuition for what this means, consider a classic example, in which nodes represent guests at a party. Color the edge between any two guests red if they’ve met before, and blue if they haven’t. You can pick any clique size you like — invite enough people to the party, and you can’t avoid either inviting a group of people who all know one another (a clique in multiple senses of the word) or inviting a group of people who have never met before.

“The simplest thing you can have in a graph is a monochromatic clique,” said Maria Chudnovsky , a mathematician at Princeton University. “It’s kind of amazing that in every huge graph you can find a big one of those. It’s completely not clear.”

The first few Ramsey numbers are relatively simple to calculate. Let’s say you want to know the size of the smallest graph that must unavoidably hold a clique of size two, or R(2) to mathematicians. Since a complete graph with two nodes is just two nodes connected by an edge, and that edge has to be either red or blue, R(2) is 2. More generally, R( k ), or the Ramsey number of k , is the minimum number of nodes beyond which a graph can’t avoid containing a clique of size k .

Samuel Velasco/ Quanta Magazine

It isn’t so hard to show that the Ramsey number for a clique of size 3, or R(3), is 6 (see graphic). But it wasn’t until 1955 that R(4) was pinned down at 18. R(5) remains unknown — it’s at least 43 and no bigger than 48. Though these numbers are small, sifting through all of the possible colorings is out of the question, said David Conlon of the California Institute of Technology. Consider the number of colorings on a complete graph with 43 nodes. “You have 903 edges; each of those can be colored in two ways,” he explained. “So you get 2 903 , which is just astronomically large.”

As the size of the clique increases, the task of nailing down the Ramsey number only gets more difficult. Erdős quipped that all-out war with mathematically demanding aliens would be easier than trying to figure out R(6) , which is somewhere between 102 and 165. The range of uncertainty grows quickly: According to estimates compiled by Stanisław Radziszowski , R(10) could be as small as 798 and as big as 23,556. But mathematicians’ goals reach far beyond the Ramsey number of 10. They want a formula that will give a good estimate of R( k ), even — or especially — when k is extremely large.

“I don’t know of a person in combinatorics who has not thought about this problem at least a little bit,” Wigderson said. “This problem is, I think, really special.”

Order in the Court

Frank Ramsey was an eclectic, brilliant figure who died when he was 26 years old. Just weeks before he died, the Proceedings of the London Mathematical Society published the paper in which he introduced Ramsey numbers. That work wasn’t even about graphs, but about a problem in mathematical logic. Ramsey proved that a statement that satisfies certain conditions has to be true at least some of the time. One of those conditions was that there be a large “universe” of scenarios to test the statement in. As a steppingstone to this result, Ramsey showed that the Ramsey number is finite.

Five years later, Erdős and Szekeres showed that the Ramsey number of k is less than 4 k . And 12 years after that, Erdős showed that it is bigger than about $latex \sqrt{2}^k$. To do that, he calculated the chances that a graph with randomly colored edges contains a monochromatic clique. This “probabilistic” technique became massively influential in graph theory. It also trapped R( k ) between two exponentially growing goalposts: $latex \sqrt{2}^k$ and $latex 4^k$.

As the decades slid by, numerous mathematicians attempted to narrow the gap between the possible values of the Ramsey number. Some succeeded: In 1975, Joel Spencer doubled the lower limit . And a series of papers by Conlon , Andrew Thomason and Ashwin Sah pushed down the upper limit by a factor of about $latex 4^{log(k)^2}$ by 2020. But compared to the sizes of the bounds on the Ramsey number, these adjustments were small. By contrast, any reduction to the 4 in Erdős and Szekeres’ formula R( k ) < 4 k would be an exponential improvement, growing quickly as k gets bigger.

Frank Ramsey (left) died before the theory bearing his name could be developed by Paul Erdős, among others.

Courtesy of Stephen Burch (left); George Csicsery

“It seems like just a cute little question,” said Rob Morris , a mathematician at IMPA, Brazil’s Institute for Pure and Applied Mathematics, in Rio de Janeiro, who co-authored the new result with Campos, Griffiths and Sahasrabudhe. “It’s a little subtle to appreciate. But people really care about it.” This is possibly an understatement. “Had they proved it in 1936, people would have said, OK, so what’s the big deal?” said Béla Bollobás, who was Morris and Sahasrabudhe’s doctoral adviser at the University of Memphis. “Since then, it has been proved that it’s a very big problem, because over the years, several thousand papers have been written on various variants of the Ramsey problem.” As Liana Yepremyan , a mathematician at Emory University, said, “The Ramsey numbers create that bridge between combinatorics and probability and geometry.”

Game Theory

  In August 2018, Sahasrabudhe was a postdoctoral fellow under Morris at IMPA. The two were hoping to start a new project with Griffiths, who teaches at the nearby Pontifical Catholic University, when a paper by Conlon caught their attention. The paper outlined a possible strategy for getting an exponential improvement on the Ramsey number. Griffiths, Morris and Sahasrabudhe began playing with the idea.

“It was really exciting in the beginning,” Sahasrabudhe recalled. It only took them about a month, he said, to lay out a sketch of their argument.

Their plan was to build on ideas used in Erdős and Szekeres’ original proof that $latex R(k) < 4^k$. To prove that the Ramsey number is at most  $latex 4^k$, imagine playing a game on a complete graph with $latex 4^k$ nodes. The game has two players. First, your opponent colors each edge either red or blue, hoping to color the edges in a way that avoids creating a monochromatic clique of k nodes.

Once your opponent is done coloring, it’s your job to search for a monochromatic clique. If you find one, you win.

To win this game, you can follow a simple strategy. It helps to think (metaphorically) about sorting your nodes into two buckets. The nodes in one bucket will form a blue clique, and the nodes in the other will form a red clique. Some nodes will be deleted, never to be heard from again. At the beginning, both buckets are empty, and every node is a candidate to go into either one.

“It seems like just a cute little question,” said Rob Morris. “It’s a little subtle to appreciate. But people really care about it.”

Divulgação IMPA

Start with any node that strikes your fancy. Then look at the connecting edges. If half or more of the edges are red, then delete all the blue edges and the nodes they are connected to. Then put your original choice into the “red” bucket. All of this node’s red edges are still alive and well, clinging onto the rest of the graph from inside the bucket. If more than half of the edges are blue, you analogously delete the red edges and nodes, and put it in the blue bucket.

Repeat until you have no nodes left. (Since the graph is complete, every remaining node at any point is connected to both buckets until it is placed into one of them.)

When you’re done, look inside the buckets. Because a node went into the red bucket only after its blue neighbors were eliminated, all the nodes in the red bucket are connected by red edges — they form a red clique. Likewise, the blue bucket forms a blue clique. If your original graph has at least $latex 4^k$ nodes, it’s possible to prove that one of these buckets must contain at least k nodes, guaranteeing a monochromatic clique in the original graph.

This argument is clever and elegant, but it has you building two cliques — one blue and one red — even though you only really need one. It would be more efficient to always go red, Conlon explained. Under this strategy, you’d pick a node at each step, erase its blue edges, and throw it into the red bucket. The red bucket would then quickly fill up, and you’d amass a red clique of k nodes in half the time.

But your strategy has to work for any red-blue coloring, and it’s hard to know whether you can always find a node with lots of red edges. So following Conlon’s suggestion runs the risk of running into a node that has almost no red edges attached to it. That would force you to delete a huge portion of the graph at once, leaving you scrambling to build your clique before you run out of nodes. To carry out Conlon’s suggestion, Griffiths, Morris and Sahasrabudhe needed to prove that this risk was avoidable.

An Open-Book Exam

In updating their gameplay, Griffiths, Morris and Sahasrabudhe followed a slightly more circuitous route. Rather than build a monochromatic clique directly by tossing nodes in their red and blue buckets, they first focused on building a structure called a red book.

A book, in graph theory, is made up of two parts: There’s a monochromatic clique, called the “spine,” and a second, distinct cluster of nodes called the “pages.” In a red book, all of the edges connecting nodes within the spine are red, as are the edges linking the spine to the pages. The edges connecting nodes within the pages, however, can be any combination of colors. Conlon had noted in his 2018 paper that if you can find a red clique inside the pages of a book, then you could combine it with the spine to get a larger red clique. This lets you decompose a search for a large red clique into two easier searches. First, look for a red book. Then look for a clique in the pages of the book.

Merrill Sherman/ Quanta Magazine; source: https://doi.org/10.48550/arXiv.2303.09521

Griffiths, Morris and Sahasrabudhe wanted to adjust the game-winning algorithm so that it built a red book instead of a red clique. Though they settled on this plan just weeks into their project, it would take years to get it to work. They still needed to stave off the threat of losing all their red edges.

“We were stuck for a very long time,” said Campos, who joined the project in 2021.

This January, the four mathematicians agreed to switch to another version of the problem. That version sounds more complicated, but it turned out to be easier.

All along, the team had been focused on the Ramsey number R( k ), also known as the “diagonal” Ramsey number. A graph of size R( k ) must contain k nodes, all connected by edges of the same color, but it doesn’t matter if that color is red or blue. On the other hand, the “off-diagonal” Ramsey number R( k , l ) measures how big a graph must be before it contains either a red clique with k nodes, or a blue clique with l nodes. Instead of continuing to hack away at the diagonal problem, the group decided to try the off-diagonal version. This proved revelatory.

“For a long time, it felt like every door you pushed on was either bolted shut, or at least pretty difficult to get through,” Griffiths said. “And after that change, you just felt like every door was open. Somehow, everything just seemed to work.” In the off-diagonal version, they found a way to delete a bunch of blue edges at once following a particular protocol, which increased the density of red edges, and led to an improved bound on the off-diagonal Ramsey number. This method, called a “density increment” argument, had previously been used to solve other important problems in combinatorics , but it had not been used on the Ramsey number problem.

The four mathematicians then used the new bound on the off-diagonal Ramsey number to clear the way for the diagonal result. By the beginning of February, they had finally lowered the limit on the Ramsey number by an exponential factor, an achievement mathematicians had sought for nearly a century. And they did it by modernizing the same line of argument that Erdős and Szekeres had put forth in 1935.

“For a long time, it felt like every door you pushed on was either bolted shut, or at least pretty difficult to get through,” said Simon Griffiths .

Francisco Muniz

Epsilon, Epsilon

After the talks on March 16, attendees began to confirm the rumors. Photos from Sahasrabudhe’s talk circulated through phone calls and private messages — even in a vague but suggestive post on the mathematician Gil Kalai’s blog. Campos, Griffiths, Sahasrabudhe and Morris claimed to have shown that $latex R(k) < 3.993^k$. That night, the four authors posted their paper online , allowing mathematicians to see the new proof for themselves.

“I think many of us did not expect to see such an improvement in our lifetime, essentially,” said Matija Bucić , a mathematician at Princeton University and the Institute for Advanced Study. “I think it’s an absolutely amazing result.”

Many experts are hopeful that, with the exponential barrier felled, more progress will quickly follow. The authors of the new paper intentionally did not push their method to its limits, to avoid clouding their argument with unnecessary details. “I’m very interested in how far the method can actually go, because I have no idea,” Campos said.

“It’s an utterly ingenious, absolutely wonderful proof, and a genuine breakthrough. So now I expect the floodgates to be opened,” Bollobás said. “I am convinced that in three years’ time, the debate will be about possible improvements. Can we improve 3.993 to 3.9? Maybe to 3.4? And what about 3?”

The new proof comes in at 56 pages, and it will take weeks for every detail to be fully verified by the combinatorics community. But colleagues are optimistic. “This group of authors, they’re very serious people. And they’re people who are really, really good at very technical things,” Wigderson said.

When it comes to his collaborators, Griffiths agrees. “It’s a privilege to work with brilliant people, isn’t it? And I think that’s what I feel very grateful for,” he said. “If they’d left it to me, I might have taken another five years to get the details right.”

Get highlights of the most important news delivered to your email inbox

Also in Mathematics

Mathematicians Prove Hawking Wrong About the Most Extreme Black Holes

The Geometric Tool That Solved Einstein’s Relativity Problem

Grad Students Find Inevitable Patterns in Big Sets of Numbers

Comment on this article.

Quanta Magazine moderates comments to facilitate an informed, substantive, civil conversation. Abusive, profane, self-promotional, misleading, incoherent or off-topic comments will be rejected. Moderators are staffed during regular business hours (New York time) and can only accept comments written in English. 

Next article

Use your social network.

Forgot your password ?

We’ll email you instructions to reset your password

Enter your new password

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

• View all journals
• Explore content
• About the journal
• Publish with us
• Sign up for alerts
• News & Views
• Published: 18 December 2023

Complex systems

Graph theory captures hard-core exclusion

• Zoltán Toroczkai   ORCID: orcid.org/0000-0002-6602-2849 1  

Nature Physics volume  20 ,  pages 20–21 ( 2024 ) Cite this article

784 Accesses

2 Altmetric

Metrics details

• Applied mathematics
• Complex networks

Physical networks, composed of nodes and links that occupy a spatial volume, are hard to study with conventional techniques. A meta-graph approach that elucidates the impact of physicality on network structure has now been introduced.

This is a preview of subscription content, access via your institution

Access options

Access Nature and 54 other Nature Portfolio journals

Get Nature+, our best-value online-access subscription

24,99 € / 30 days

cancel any time

Subscribe to this journal

Receive 12 print issues and online access

195,33 € per year

only 16,28 € per issue

Buy this article

• Purchase on SpringerLink
• Instant access to full article PDF

Prices may be subject to local taxes which are calculated during checkout

Barabási, A.-L. Network Science (Cambridge University Press, 2016).

Wasserman, S. & Faust, K. Social Network Analysis: Methods and Applications (Cambridge University Press, 1994).

Ecological Networks: Linking Structure to Dynamics in Food Webs (eds Pascual, M. & Dunne, J. A.) (Oxford University Press, 2005).

Pastore y Piontti, A. et al. Charting the Next Pandemic: Modeling Infectious Disease Spreading in the Data Science Age (Springer, 2019).

Pósfai, M. et al. Nat. Phys. https://doi.org/10.1038/s41567-023-02267-1 (2023).

Article   Google Scholar  

Download references

Author information

Authors and affiliations.

Department of Physics and Astronomy, University of Notre Dame, Notre Dame, IN, USA

Zoltán Toroczkai

You can also search for this author in PubMed   Google Scholar

Corresponding author

Correspondence to Zoltán Toroczkai .

Ethics declarations

Competing interests.

The author declares no competing interests.

Rights and permissions

Reprints and permissions

About this article

Cite this article.

Toroczkai, Z. Graph theory captures hard-core exclusion. Nat. Phys. 20 , 20–21 (2024). https://doi.org/10.1038/s41567-023-02342-7

Download citation

Published : 18 December 2023

Issue Date : January 2024

DOI : https://doi.org/10.1038/s41567-023-02342-7

Share this article

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Quick links

• Explore articles by subject
• Guide to authors
• Editorial policies

Sign up for the Nature Briefing: AI and Robotics newsletter — what matters in AI and robotics research, free to your inbox weekly.

Breadcrumbs Section. Click here to navigate to respective pages.

Recent Advancements in Graph Theory

DOI link for Recent Advancements in Graph Theory

Get Citation

Graph Theory is a branch of discrete mathematics. It has many applications to many different areas of Science and Engineering. This book provides the most up-to-date research findings and applications in Graph Theory.

This book focuses on the latest research in Graph Theory. It provides recent findings that are occurring in the field, offers insights on an international and transnational levels, identifies the gaps in the results, and includes forthcoming international studies and research, along with its applications in Networking, Computer Science, Chemistry, and Biological Sciences, etc.

The book is written with researchers and post graduate students in mind.

TABLE OF CONTENTS

Chapter 1 | 8  pages, graceful labeling for eight sprocket graph, chapter 2 | 10  pages, universal absolute mean graceful graphs, chapter 3 | 6  pages, universal α-graceful gear related graphs, chapter 4 | 9  pages, l(2, 1) - labeling on jahangir graph, chapter 5 | 11  pages, v 4 - cordial labeling of some ladder and book related graphs, chapter 6 | 9  pages, sd-prime cordial labeling of double k-polygonal snake graph, chapter 7 | 17  pages, edge product cordial and total edge product cordial labeling of some wheel related graphs, chapter 8 | 8  pages, product cordial labeling for the line graph of bistar, chapter 9 | 16  pages, sum divisor cordial labeling for vertex switching of cycle related graphs, chapter 10 | 15  pages, a few results on fibonacci cordial labeling, chapter 11 | 17  pages, some more parity combination cordial graphs, chapter 12 | 13  pages, total neighborhood prime labeling of join graphs, chapter 13 | 15  pages, gaussian vertex prime labeling of some graphs obtained from origami models, chapter 14 | 10  pages, vertex magic total labeling of tensor product of cycles, chapter 15 | 8  pages, antimagic labeling of some star and bistar related graphs, chapter 16 | 11  pages, distance magic and distance antimagic labeling of some product graphs, chapter 17 | 13  pages, graphs from subgraphs, chapter 18 | 13  pages, unit graphs having their domination number half their order, chapter 19 | 11  pages, the pendant number of some graph products, chapter 20 | 13  pages, wiener index of tensor product of cycle graph and some other graphs, chapter 21 | 14  pages, wiener index of some zero-divisor graphs, chapter 22 | 11  pages, algebraic signed graphs: a review, chapter 23 | 6  pages, nullity and energy of complete tripartite graphs, chapter 24 | 10  pages, some new results on restrained edge domination number of graphs, chapter 25 | 26  pages, some new graph coloring problems, chapter 26 | 13  pages, total global dominator coloring of graphs, chapter 27 | 10  pages, rainbow vertex connection number of a class of triangular snake graph, chapter 28 | 14  pages, hamiltonian chromatic number of trees, chapter 29 | 7  pages, some results on degree sum energy of a graph, chapter 30 | 20  pages, randić energy of some graphs, chapter 31 | 10  pages, l-spectra of graphs obtained by duplicating graphs elements.

• Privacy Policy
• Terms & Conditions
• Cookie Policy
• Taylor & Francis Online
• Taylor & Francis Group
• Students/Researchers
• Librarians/Institutions

Connect with us

Registered in England & Wales No. 3099067 5 Howick Place | London | SW1P 1WG © 2024 Informa UK Limited

Information

• Author Services

Initiatives

You are accessing a machine-readable page. In order to be human-readable, please install an RSS reader.

All articles published by MDPI are made immediately available worldwide under an open access license. No special permission is required to reuse all or part of the article published by MDPI, including figures and tables. For articles published under an open access Creative Common CC BY license, any part of the article may be reused without permission provided that the original article is clearly cited. For more information, please refer to https://www.mdpi.com/openaccess .

Feature papers represent the most advanced research with significant potential for high impact in the field. A Feature Paper should be a substantial original Article that involves several techniques or approaches, provides an outlook for future research directions and describes possible research applications.

Feature papers are submitted upon individual invitation or recommendation by the scientific editors and must receive positive feedback from the reviewers.

Editor’s Choice articles are based on recommendations by the scientific editors of MDPI journals from around the world. Editors select a small number of articles recently published in the journal that they believe will be particularly interesting to readers, or important in the respective research area. The aim is to provide a snapshot of some of the most exciting work published in the various research areas of the journal.

Original Submission Date Received: .

• Active Journals
• Find a Journal
• Proceedings Series
• For Authors
• For Reviewers
• For Editors
• For Librarians
• For Publishers
• For Societies
• For Conference Organizers
• Open Access Policy
• Institutional Open Access Program
• Special Issues Guidelines
• Editorial Process
• Research and Publication Ethics
• Article Processing Charges
• Testimonials
• Preprints.org
• SciProfiles
• Encyclopedia

Journal Menu

• Mathematics Home
• Aims & Scope
• Editorial Board
• Reviewer Board
• Topical Advisory Panel
• Instructions for Authors
• Special Issues
• Sections & Collections
• Article Processing Charge
• Indexing & Archiving
• Editor’s Choice Articles
• Most Cited & Viewed
• Journal Statistics
• Journal History
• Journal Awards
• Society Collaborations
• Conferences
• Editorial Office

Journal Browser

• arrow_forward_ios Forthcoming issue arrow_forward_ios Current issue
• Vol. 12 (2024)
• Vol. 11 (2023)
• Vol. 10 (2022)
• Vol. 9 (2021)
• Vol. 8 (2020)
• Vol. 7 (2019)
• Vol. 6 (2018)
• Vol. 5 (2017)
• Vol. 4 (2016)
• Vol. 3 (2015)
• Vol. 2 (2014)
• Vol. 1 (2013)

Find support for a specific problem in the support section of our website.

Please let us know what you think of our products and services.

Visit our dedicated information section to learn more about MDPI.

Graph-Theoretic Problems and Their New Applications

• Print Special Issue Flyer
• Special Issue Editors

Special Issue Information

Benefits of publishing in a special issue.

• Published Papers

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section " Mathematics and Computer Science ".

Deadline for manuscript submissions: closed (31 January 2020) | Viewed by 69652

Share This Special Issue

Special issue editor.

Dear Colleagues,

Nowadays, graph theory plays a vital role in many disciplines. We invite you to submit your latest original research results in all aspects of graph theory to this Special Issue. We are looking both for new and innovative approaches for traditional graph-theoretic problems and well as for new applications of graph theory in emerging fields, such as in network security, computer science and data analysis, bioinformatics, operations research, engineering and manufacturing, physics and chemistry, linguistics, or social sciences. Both theoretical results, as well as new graph algorithms, with investigations of their computational complexity will be considered. Below, we give an exemplary, but not exhaustive, list of potential subjects for submissions to this Special Issue.

Prof. Dr. Frank Werner Guest Editor

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website . Once you are registered, click here to go to the submission form . Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

• Deterministic, randomized, exact and approximative graph algorithms
• Paths, cycles, and trees
• Network theory
• Graph labeling
• Graph coloring
• Symmetric graphs
• Polyhedral graphs
• Topological indices
• Domination in graphs
• Applications of graph theory
• Ease of navigation: Grouping papers by topic helps scholars navigate broad scope journals more efficiently.
• Greater discoverability: Special Issues support the reach and impact of scientific research. Articles in Special Issues are more discoverable and cited more frequently.
• Expansion of research network: Special Issues facilitate connections among authors, fostering scientific collaborations.
• External promotion: Articles in Special Issues are often promoted through the journal's social media, increasing their visibility.
• e-Book format: Special Issues with more than 10 articles can be published as dedicated e-books, ensuring wide and rapid dissemination.

Further information on MDPI's Special Issue polices can be found here .

Published Papers (21 papers)

Jump to: Research , Review

Jump to: Editorial , Review

Jump to: Editorial , Research

Further Information

Mdpi initiatives, follow mdpi.

Subscribe to receive issue release notifications and newsletters from MDPI journals

arXiv's Accessibility Forum starts next month!

Help | Advanced Search

Computer Science > Data Structures and Algorithms

Title: graph theory and its uses in graph algorithms and beyond.

Abstract: Graphs are fundamental objects that find widespread applications across computer science and beyond. Graph Theory has yielded deep insights about structural properties of various families of graphs, which are leveraged in the design and analysis of algorithms for graph optimization problems and other computational optimization problems. These insights have also proved helpful in understanding the limits of efficient computation by providing constructions of hard problem instances. At the same time, algorithmic tools and techniques provide a fresh perspective on graph theoretic problems, often leading to novel discoveries. In this thesis, we exploit this symbiotic relationship between graph theory and algorithms for graph optimization problems and beyond. This thesis consists of three parts. In the first part, we study a graph routing problem called the Node-Disjoint Paths (NDP) problem. Given a graph and a set of source-destination pairs of its vertices, the goal is to route the maximum number of pairs via node-disjoint paths. We come close to resolving the approximability of NDP by showing that it is $n^{\Omega(1/poly\log\log n)}$-hard to approximate, even on grid graphs, where n is the number of vertices. In the second part of this thesis, we use graph decomposition techniques developed for efficient algorithms to derive a graph theoretic result. We show that for every n-vertex expander graph G, if H is any graph with at most $O(n/\log n)$ vertices and edges, then H is a minor of G. In the last part, we show that the graph theoretic tools and graph algorithmic techniques can shed light on problems seemingly unrelated to graphs. We show that the randomized space complexity of the Longest Increasing Subsequence (LIS) problem in the streaming model is intrinsically tied to the query-complexity of the Non-Crossing Matching problem on graphs in a new model of computation that we define.

Submission history

Access paper:.

• Other Formats

References & Citations

• Google Scholar
• Semantic Scholar

BibTeX formatted citation

Bibliographic and Citation Tools

Code, data and media associated with this article, recommenders and search tools.

• Institution

arXivLabs: experimental projects with community collaborators

arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website.

Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them.

Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs .