#graphtheory

Graphs & conjectures:
#mathematics #graphtheory #networks #research #science

https://www.quantamagazine.org/new-proof-settles-decades-old-bet-about-connected-networks-20250418/

#git can be hard, like anything if you want to understand it
So are DAGs
Then again of you want to start with git and get to #graphtheory , which is fun in my view. It ll be after months of accidents
No employer wants that on their payroll and no team or authority wants to be responsible for it or accepts it and it ll be a nightmare for the person after fun.
So the society, the institution and the market collectively orient workforce not to have fun in learning things, including cubicles.
Oh yes move fast and break thing , but at your expense, which clearly is hoarded.
That explains a lot , including the rise of #ai
#git
https://ohshitgit.com/

@paysmaths
Lowell Beineke, still attending graph theory talks in April 2025 (seated on the right)
#math #GraphTheory #PurdueFortWayne

Ok, someone more familiar with graph theory tell me how to compute a minimum weight perfect matching on a complete graph with even-number of vertices? Or at least point me to a resource?

It's all either left as an exercise to the reader, extremely complex algorithms (Blossom or something?), or weird libraries, where apparently if I use it on complex graphs it's not necessary, but they then point me to even more complex papers.

Thought I had it solved but now it's returning a matching that's too small.

#graphtheory #computerscience #academia

Without formal training, the way I modeled physical distribution system #Infrastructure for utilities districts in my #Django app evolved significantly over time.

As I learned more about graph theory and simulation, I figured out how to model for both the physical structure and the logical aspects of their interconnection and behavior in various scenarios.

#GraphTheory #WaterInfrastructure #Modeling

1/6

For various (mathematical, meteorological, alimentary) reasons, I usually prefer 2π day.
Nevertheless, today I make the following offering:

http://arxiv.org/abs/2503.10002

Pjotr Buys, @Janvadehe and I used Shearer's induction to address the question:

How few independent sets can a triangle-free graph of average degree d have?

The answer is close to how many a random graph has.
What is perhaps surprising is just *how* close it comes.
(I queried the combinatorial hive mind about this last week.)

#combinatorics #graphtheory #ExtremalCombinatorics #probability #math #mathematics #piDay

This video is a really pretty visualization of the A* pathfinding algorithm using #OpenStreetMap road network data for #Chicago and #Rome as examples.

https://youtu.be/CgW0HPHqFE8

#ComputerScience #algorithms #GraphTheory #dataviz

Cactus Language • Overview 3.2
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/

Given a body of conceivable propositions we need a way to follow the threads of their indications from their object domain to their values for the mind and a way to follow those same threads back again. Moreover, we need to implement both ways of proceeding in computational form. Thus we need programs for tracing the clues sentences provide from the universe of their objects to the signs of their values and, in turn, from signs to objects. Ultimately, we need to render propositions so functional as indicators of sets and so essential for examining the equality of sets as to give a rule for the practical conceivability of sets. Tackling that task requires us to introduce a number of new definitions and a collection of additional notational devices, to which we now turn.

Resources —

Cactus Language • Overview
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Overview

Survey of Animated Logical Graphs
• https://inquiryintoinquiry.com/2024/03/18/survey-of-animated-logical-graphs-7/

Survey of Theme One Program
• https://inquiryintoinquiry.com/2024/02/26/survey-of-theme-one-program-6/

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory

Cactus Language • Overview 3.1
• https://inquiryintoinquiry.com/2025/03/07/cactus-language-overview-3/

In the development of Cactus Language to date the following two species of graphs have been instrumental.

• Painted And Rooted Cacti (PARCAI).
• Painted And Rooted Conifers (PARCOI).

It suffices to begin with the first class of data structures, developing their properties and uses in full, leaving discussion of the latter class to a part of the project where their distinctive features are key to developments at that stage. Partly because the two species are so closely related and partly for the sake of brevity, we'll always use the genus name “PARC” to denote the corresponding cacti.

To provide a computational middle ground between sentences seen as syntactic strings and propositions seen as indicator functions the language designer must not only supply a medium for the expression of propositions but also link the assertion of sentences to a means for inverting the indicator functions, that is, for computing the “fibers” or “inverse images” of the propositions.

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#AutomataTheory #FormalLanguages #FormalGrammars #GraphTheory

Cactus Language • Overview 1.2
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/

Resource —

For readers interested and intrepid enough to read ahead, here’s an outline of my work in progress on the OEIS Wiki, which I’ll be revising and serializing to my Inquiry blog.

Part 1
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1

Cactus Language • Syntax
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_1#Syntax

Part 2
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2

Generalities About Formal Grammars
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_2#Generalities

Part 3
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3

Cactus Language • Stylistics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stylistics

Cactus Language • Mechanics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Mechanics

Cactus Language • Semantics
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Semantics

Stretching Exercises
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Part_3#Stretching_Exercises

References
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_References

Document History
• https://oeis.org/wiki/Cactus_Language_%E2%80%A2_Document_History

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory

Cactus Language • Overview 1.1
• https://inquiryintoinquiry.com/2025/03/01/cactus-language-overview-1/

❝Thus, what looks to us like a sphere of scientific knowledge more accurately should be represented as the inside of a highly irregular and spiky object, like a pincushion or porcupine, with very sharp extensions in certain directions, and virtually no knowledge in immediately adjacent areas. If our intellectual gaze could shift slightly, it would alter each quill’s direction, and suddenly our entire reality would change.❞

— Herbert J. Bernstein • “Idols of Modern Science”

The following report describes a calculus for representing propositions as sentences, that is, as syntactically defined sequences of signs, and for working with those sentences in light of their semantically defined contents as logical propositions. In their computational representation the expressions of the calculus parse into a class of graph‑theoretic data structures whose underlying graphs are called “painted cacti”.

Painted cacti are a specialization of what graph‑theorists refer to as “cacti”, which are in turn a generalization of what they call “trees”. The data structures corresponding to painted cacti have especially nice properties, not only useful in computational terms but interesting from a theoretical standpoint. The remainder of the present Overview is devoted to motivating the development of the indicated family of formal languages, going under the generic name of Cactus Language.

#Peirce #Logic #Semiotics #LogicalGraphs #DifferentialLogic
#Automata #FormalLanguages #FormalGrammars #GraphTheory

A question for the (combinatorial) hive mind.

There are a lot of extremal results that are matched asymptotically by some probabilistic construction, but with some gap, often quite substantial. I'm thinking about the Ramsey numbers R(k,k) or R(3,k), but examples of this phenomenon are prevalent.

I'm curious, does someone out there know of good examples of (extremal) results where some probabilistic construction (e.g. via a random graph) is matched asymptotically, and very precisely?

#graphtheory #combinatorics

Visualizing Interconnected Networks with Matplotlib and NetworkX
Learn to visualize complex networks using Network Visualization Matplotlib & NetworkX in Python. Create insightful visualizations & communicate complex data clearly! #NetworkVisualization #Python #Matplotlib #NetworkX #DataVisualization #GraphTheory
https://tech-champion.com/programming/python-programming/visualizing-interconnected-networks-with-matplotlib-and-networkx/

Network Graph Visualization: Improving Clarity in Dense Clusters with NetworkX
Improve Network Graph Visualization with NetworkX & Matplotlib! Learn simple yet effective strategies to enhance clarity, especially in dense clusters. Explore alternative layout algorithms & parameter adjustments for insightful data representation. #NetworkGraphVisualization #NetworkX #Matplotlib #DataVisualization #GraphTheory #Python
https://tech-champion.com/programming/python-programming/network-graph-visualization-improving-clarity-in-dense-clusters-with-networkx/

Refined my haiku visual a little today. I think it would be fun to see if there are any paths to take these words and traverse all of the haiku and if so - how many paths.

The dataset is Creative Commons if anyone wants to fork and play.

https://codepen.io/fractalkitty/pen/QwWbGjd

#recurse #data #haiku #poetry #datavis #graphTheory

Starting out in mathematical research, especially in discrete mathematics, a big focus is problem-solving. It's like a race, and once you've solved one, you set out right away for the next adrenaline rush.

Take for granted a bustling market of open problems (again, especially in discrete mathematics). Scour papers or problem sites. Challenge close colleagues with the ones that eluded you. The harder, the better, right? There is occasionally awkward coffee talk of that intangible `taste' or `judgement', but, come on, less talk and more solving!

(please imagine here a subtly ironic tone in my voice)

(1/3)

#math
#graphtheory
#combinatorics
#ExtremalCombinatorics

A post of @11011110 has reminded me that (after a year and a half lurking here) it's never too late for me to toot and pin an intro here.

I am a Canadian mathematician in the Netherlands, and I have been based at the University of Amsterdam since 2022. I also have some rich and longstanding ties to the UK, France, and Japan.

My interests are somewhere in the nexus of Combinatorics, Probability, and Algorithms. Specifically, I like graph colouring, random graphs, and probabilistic/extremal combinatorics. I have an appreciation for randomised algorithms, graph structure theory, and discrete geometry.

Around 2020, I began taking a more active role in the community, especially in efforts towards improved fairness and openness in science. I am proud to be part of a team that founded the journal, Innovations in Graph Theory (https://igt.centre-mersenne.org/), that launched in 2023. (That is probably the main reason I joined mathstodon!) I have also been a coordinator since 2020 of the informal research network, A Sparse (Graphs) Coalition (https://sparse-graphs.mimuw.edu.pl/), devoted to online collaborative workshops. In 2024, I helped spearhead the MathOA Diamond Open Access Stimulus Fund (https://www.mathoa.org/diamond-open-access-stimulus-fund/).

Until now, my posts have mostly been about scientific publishing and combinatorics.

#introduction
#openscience
#diamondopenaccess
#scientificpublishing
#openaccess
#RemoteConferences
#combinatorics
#graphtheory
#ExtremalCombinatorics
#probability

The Hidden Networks
That Rule Our World
S2 E50

Join us for a fascinating deep dive into the world of network analysis, where we explore Node2Vec - a groundbreaking algorithm that helps us understand the hidden communities within complex networks. Unlock profound insights about communities hidden within the vast connections surrounding us.

https://helioxpodcast.substack.com/p/the-hidden-networks-that-rule-our

#NetworkAnalysis #AI #ComplexSystems #GraphTheory #Node2Vec #NeuralNetworks #SocialNetworks #Podcast

https://www.buzzsprout.com/2405788/episodes/16420355

Riffs and Rotes • Happy New Year 2025
• https://inquiryintoinquiry.com/2025/01/01/riffs-and-rotes-happy-new-year-2025/

\( \text{Let} ~ p_n = \text{the} ~ n^\text{th} ~ \text{prime}. \)

\( \text{Then} ~ 2025
= 81 \cdot 25
= 3^4 5^2 \)

\( = {p_2}^4 {p_3}^2
= {p_2}^{{p_1}^{p_1}} {p_3}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_2}}^{p_1}
= {p_{p_1}}^{{p_1}^{p_1}} {p_{p_{p_1}}}^{p_1} \)

No information is lost by dropping the terminal 1s. Thus we may write the following form.

\[ 2025 = {p_p}^{p^p} {p_{p_p}}^p \]

The article linked below tells how forms of that sort correspond to a family of digraphs called “riffs” and a family of graphs called “rotes”. The riff and rote for 2025 are shown in the next two Figures.

Riff 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/riff-2025.png

Rote 2025
• https://inquiryintoinquiry.files.wordpress.com/2025/01/rote-2025.png

Reference —

Riffs and Rotes
• https://oeis.org/wiki/Riffs_and_Rotes

#Arithmetic #Combinatorics #Computation #Factorization #GraphTheory #GroupTheory
#Logic #Mathematics #NumberTheory #Primes #Recursion #Representation #RiffsAndRotes

I'm doing some of my "follow the instructions for a change" #Sashiko, and came upon a bit that I think means I can't do everything that's left without a lot of stopping and starting.

Then I realised I have an actual graph theorist in the house who might see something I'd missed.

So I checked with @ColinTheMathmo, and he did spot something I'd missed, but it confirmed the impossibility.

I'll have to fudge it...

#MathsChat #GraphTheory #HandEmbroidery