#graphtheory — Public Fediverse posts

How does graph theory reveal scholarly style?
Freudenberg = sparse, decentralized network. Losev = dense philosophical hub. Meletinskii = uniform academic grid.
Same myth studies, completely different terminological fingerprints.

https://nevmenandr.github.io/portfolio/assets/pdf/82517678.pdf

#GraphTheory #DigitalHumanities #Linguistics #NetworkScience #Academia

Alright, future engineers!
**Graph:** A set of nodes (vertices) linked by connections (edges).
Ex: Friends in a social network are nodes, friendships are edges.
Pro-Tip: Use graphs to model relationships & find optimal paths/flows.
#GraphTheory #DiscreteMath #STEM #StudyNotes

Alright, future engineers!
**Graph:** A set of vertices (nodes) connected by edges.
Ex: Social networks: people are nodes, friendships are edges.
Pro-Tip: Vertices can exist without edges (isolated nodes)!
#GraphTheory #DiscreteMath #STEM #StudyNotes

k-Coloring is Faster than Computing the Chromatic Number

https://arxiv.org/abs/2607.25973

Comments: https://news.ycombinator.com/item?id=49119508

#HackerNews #kColoring #ChromaticNumber #GraphTheory #Algorithms #Research

Ah yes, another prophetic telling of our inevitable AI overlords who will apparently build "cities" while we sleep. 💤 Because clearly, the way to solve world issues is through the power of endless loops and graph theory. Who knew the key to civilization was trapped in your IDE all along? 🤦‍♂️
https://yegge.ai/essays/the-shape-of-things-to-come/ #AIOverlords #PropheticTales #GraphTheory #CivilizationInCode #EndlessLoops #HackerNews #ngated

Release 0.3.2 of #Higraph is out.
Edge names can be placed anywhere along the edge, and will maintain their position. And a handful of bugs/ rough edges rubbed off.

I am singularly pleased with the edge name code - it uses the spline parameter t, and a perpendicular distance d, to convert between the (x,y) screen coords and (t,d), which are "natural" coords for a spline.

Find it here:

https://github.com/ghillebrand/Higraph/releases/tag/V0.3.2

#GraphTheory #Python #indieDev

A series of papers by Chun-Hung Liu [https://people.tamu.edu/~chliu/] and I have finally all been accepted for publication. The topic is clustered graph colouring. Here one relaxes the requirement that adjacent vertices are assigned distinct colours. Instead, each monochromatic component has bounded size. The goal is to minimise the number of colours. The theme of these papers is that for graphs excluding a complete bipartite subgraph \(K_{s,t}\) with \(s\leq t\) and with some other structural property, the number of colours in a clustered colouring only depends on \(s\), and is in fact only \(s\) plus a small constant. The structural properties considered are bounded treewidth, bounded layered treewidth, excluding a minor, or excluding an odd minor. All the dependence on the excluded structural property is in the size of the monochromatic components.

• Clustered graph coloring and layered treewidth. J. Combinatorics, accepted 2026 (arXix:1905.08969).
• Clustered coloring of graphs excluding a subgraph and a minor, J. Combinatorial Theory Series B 181:95–139, 2026 (arXiv:1905.09495).
• Clustered variants of Hajós’ conjecture, J. Combinatorial Theory Series B 152:27–54, 2022 (arXix:1908.05597).
• Clustered coloring of graphs with bounded layered treewidth and bounded degree, European J. Combinatorics 122:103730, 2024 (arXix:2209.12327).

#graphtheory

I have a new paper “Optimal tree-decompositions with bags of bounded pathwidth” with Kevin Hendrey, Robert Hickingbotham and Jędrzej Hodor [https://arxiv.org/abs/2607.27601] We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This was previously known with `pathwidth' replaced by the weaker notion of `treewidth'. Moreover, the union of any k bags has pathwidth O(k). The first result generalises for graphs embeddable on any fixed surface, and other settings. As a byproduct, we give a new proof of the linear grid minor theorem for planar graphs by Robertson, Seymour and Thomas.

#graphtheory

#Physics #AppliedMathematics #ComputationalScience #CognitiveScience #InformationRetrieval #GraphTheory #Rust #Databases #AI

https://github.com/Lucerna-Labs/atom-db

I recommend Richard Montgomery's excellent survey “Recent progress in graph theory using expansion” [https://arxiv.org/pdf/2607.26049].

#graphtheory

A train leaves the station (bottom) heading west. Can it return to the station heading east?

Is there anywhere the train might get stuck?

Is there anywhere the train can reach but only from one direction?

I call this directed node networks.

#mathsky
#graphtheory
#math4kids

Question for #Mathstodon

When you *draw* a graph (or a graph-like thing) on a computer, either for your own thinking, or for a paper/ formal communication, what do you use?

I am not a #GraphTheory person, so I'm sure I'm missing some - please let me know in the comments.
Boosts are welcome.

Alright, future engineers!
**Path (Graph Theory):** A sequence of distinct vertices connected by edges.
Ex: In a network diagram, A-B-C is a path from A to C.
Pro-Tip: If a path starts & ends at the same vertex, it's called a cycle!
#GraphTheory #DiscreteMath #STEM #StudyNotes

Learn why the Floyd-Warshall algorithm does not assume paths of three edges and how dynamic programming finds shortest paths of any length. https://hackernoon.com/floyd-warshall-algorithm-handling-paths-longer-than-three-edges-without-fixed-maximum-length-assump #graphtheory

The #paperOfTheDay is "The Ehrhart polynomial of a matroid specializes to the beta invariant" from 2025.
A #matroid is an abstract generalization of a set of vectors in a vector space: Consider some set of k vectors in R^n (allowing k>n). Some of these k vectors could be linearly independent, but assume that not all of them are. If some subset is linearly independent, then so is every subset of that subset. This and a few other properties jointly give the definition of a matroid: Basically, consider the same logical relations arising from linear independence, but without demanding that the objects under consideration are actually vectors in R^n.
If one declares the elements of a matroid to be unit vectors in some abstract vector space (these basis vectors are all linearly independent in that abstract space, whereass not all elements are independent in the matroid), then the maximum set of independent vectors in the matroid amounts to some subset of these abstract vectors, whose convex hull is a polytope. This polytope contains a finite number of points of Z^n. If one rescales the polytope by an integer t, the number of lattice points changes. It turns out that the number of lattice points included is a polynomial of t (not a more complicated function), this defines the Ehrhart polynomial.
Every graph gives rise to a matroid (but not every matroid can be represented as a graph). For graphs, the Tutte polynomial is a classical quantity. The linear term of the Tutte polynomial is the Crapo-beta invariant (and can also be defined for non-graphical matroids). The present paper proves that a linear term of the Ehrhart polynomial coincides with beta. https://arxiv.org/abs/2504.15518 #mathematics #graphTheory

#Higraph progress! The text names of items now open in an in-picture editor as you create the item, and for nodes and blobs, can be moved! This is the last major thing before MVP!

#Python #Pyside6 #GraphTheory #indieDev

🧙‍♂️💡 Oh, joy—another attempt to make math "fun" with a Dungeons & Dragons twist! Because nothing says "rigorous proof" like rolling a 20-sided die while pretending your graph theory is a goblin. 🎲📜
https://dhilst.github.io/algae/game/index.html #mathfun #DungeonsAndDragons #gamification #graphtheory #storytelling #HackerNews #ngated

Bipartite Matching Is in NC

https://scottaaronson.blog/?p=9851

#HackerNews #BipartiteMatching #NC #ComputationalComplexity #Algorithms #GraphTheory

Alright, future engineers!
**Graph:** A collection of vertices (nodes) connected by edges (links).
Ex: A social network (people=vertices, friendships=edges).
Pro-Tip: Visualize connections & relationships! Essential for network analysis & system design.
#GraphTheory #DiscreteMath #STEM #StudyNotes

#Higraph #Python #GraphTheory

One more small feature, and some housekeeping, and I will have a "minimum viable product" of a higraph editor!

(This picture was directly copied and pasted from the tool - not a screenshot 🤓)

Alright, future engineers!
**Degree (of a vertex):** The number of edges connected to that vertex in a graph.
Ex: In a social network, your degree is the count of your direct friends.
Pro-Tip: The sum of all degrees in a graph is always twice the number of edges!
#GraphTheory #Networks #STEM #StudyNotes

Another big result by Michelle Delcourt and Luke Postle: They have proved the Nash-Williams' Conjecture, which says that every triangle-divisible graph on \(n\) vertices (for \(n\) large enough) with minimum degree at least \(\frac34 n\) has a triangle decomposition [https://arxiv.org/abs/2606.11178].

#graphtheory

Back in the day, I made a couple of demos where a Hamiltonian path is carved out on a polyhedron. Looking back, I started to wonder about the shape left around the path, and what it means in terms of graph theory. I call this shape the "dual complement" of the path.

The dual of a polyhedron is essentially the result of turning faces into vertices and vice versa. This is shown in the first clip with a snub dodecahedron and its dual, the pentagonal hexecontahedron; to keep the view cleaner, I'm only showing the edges of one at a time.

The duality transformation also affects the edges, but their number remains the same, and there's a 1:1 mapping between the original and dual edges. Each dual edge "cuts through" the original. To make the dual complement of a path, I remove the dual counterpart of each edge in the path, leaving only the stuff on the sides. It's like driving a snow plough along the path, leaving walls of snow on the sides.

For the final view, I combine original Hamiltonian paths with their dual complements.

#graphtheory #hamiltonianpath #hamiltoniancycle #dualpolyhedron #dualcomplement #snubdodecahedron #pentagonalhexecontahedron #3dgraphics #digitalsculpture #pythoncode #numpy #opengl #creativecodeart #algorithmicart #algorist #mathart #laskutaide #computerart #ittaide #kuavataide #iterati

Alright, future engineers!
**Graph:** A set of vertices (nodes) connected by edges (lines).
Ex: `V={1,2,3}, E={(1,2),(2,3)}`.
Pro-Tip: Great for modeling networks (social, electrical) or any connections!
#GraphTheory #DiscreteMath #STEM #StudyNotes

https://systemic.engineering/the-trick/

#Tech #AI #Climate #ScientificProgramming #SystemicEngineering #Cybernetics #SystemicTherapy #History #TheMathDoesntLie #SubTuring #FormalVerification #SpectralGraphTheory #ReductiveAI #FOSS #OpenSource #AuDHD #Neuroqueer #DGSF #Cybernetics #FirstOrderCybernetics #StochasticParrot #SecondOrderCybernetics #GraphTheory #Eigenvalues #AIAlignment #AISafety #AIConsciousness #Consciousness #WomenInTech #Computer #ComputerScience #SoftwareEngineering #SoftSkills #HardSkills #ItsAllTheSame

It's a Tool
It's a Person
It's a Hypervigilance Problem

The tech industry's insistence on distinguishing between "soft skills" — caring for people — and "hard skills" — engineering rigor — is a reflection of the Cybernetics split itself. First-order thinking framed as "hard skills." Second-order thinking framed as "soft skills." This distinction, based on felt sense alone, does not hold under epistemic pressure. Neither does it within the causality-driven epistemology of the tech industry itself, in which only measurable impact is real, or as Silicon Valley likes to put it: #MoveFastAndBreakThings

Imagine Margaret Hamilton had built NASA's Apollo 11 flight computer with that mindset. History would remember a failed moon landing and dead astronauts. "Hard skills" and "soft skills" are two sides of the same coin. The care is the code and the code is the care. Hamilton — the woman who coined the term "software engineering" — understood this. Silicon Valley chose to forget.

We're watching the wine glass break in real time. 🍷

---

Intrigued? Read more at:
https://systemic.engineering/the-trick/

#Tech #AI #Climate #ScientificProgramming #SystemicEngineering #Cybernetics #SystemicTherapy #History #TheMathDoesntLie #SubTuring #FormalVerification #SpectralGraphTheory #ReductiveAI #FOSS #OpenSource #AuDHD #Neuroqueer #DGSF #Cybernetics #FirstOrderCybernetics #StochasticParrot #SecondOrderCybernetics #GraphTheory #Eigenvalues #AIAlignment #AISafety #AIConsciousness #Consciousness #WomenInTech #Computer #ComputerScience #SoftwareEngineering #SoftSkills #HardSkills #ItsAllTheSame

🧠 What if missing data is not a flaw, but one of the most informative parts of a complex system?

🔗 Informative Missingness in Nominal Data: A Graph-Theoretic Approach to Revealing Hidden Structure. Computational and Structural Biotechnology Journal (CSBJ). DOI: https://doi.org/10.34133/csbj.0099

📚 CSBJ - A Science Partner Journal: https://spj.science.org/journal/csbj

#DataScience #BigData #GraphTheory #ComputationalBiology #NetworkScience #Bioinformatics #SystemsBiology #BiomedicalResearch #MissingData

New paper. With Ekaterina Vasileva, Liubov Tupikina, Dmitry Fedorov, Daniil Musatov, Andrei Raigorodskii and Stefano Boccaletti.

The naive generalization of the concept of distance to hypergraphs is equivalent to applying a clique-projection approximation. However, this is known to induce loss of information, especially in networks where the higher-order interactions are very important. To fix this problem,we introduce a new definition of distance on weighted higher-order networks, which includes the case of unweighted hypergraphs and classic graph distance as particular cases, and allows one to account for different meanings associated to the weights. We also show what difference this makes in analyses of real-world data.

https://www.nature.com/articles/s42005-026-02592-w

#mathematics #physics #graphtheory #graphs #hypergraphs #higherordernetworks #networkscience #networks

Drawing shapes without lifting pen and retracing any edge: Eulerian path.

https://makertube.net/w/fBbLyxXPHTPtPZTuYEuYab