# graph theory Published articles for graph theory. This is one page of public article previews, not the complete archive. Follow Next page to continue. Summaries are not the original full articles. ## The Palindrome Announces a Graph Theory for Visual Learners Video Course DevFeed: [The Palindrome Announces a Graph Theory for Visual Learners Video Course]() Original publisher: [Read original article]() Author: Tivadar Danka Published: 2026-09-12T08:47:28Z Content type: opinion Language: en Sources: [The Palindrome]() Topics: [Graphs](), [math]() Tags: [graph](), [graph-theory](), [subscriber](), [video]() ### AI overview The Palindrome announces a planned comprehensive graph theory video course called "Graph Theory for Visual Learners" and launches a support campaign for paid subscribers and founding members. The project will use custom animations and include an upcoming video release. ### Source excerpt I'm creating The Palindrome's most ambitious video yet. Become a paid subscriber and be part of it. ## Using LLMs with SMT Solvers for Automated Reasoning and Formal Verification DevFeed: [Using LLMs with SMT Solvers for Automated Reasoning and Formal Verification]() Original publisher: [Read original article]() Published: 2026-04-02T00:00:00Z Content type: opinion Language: en Sources: [Alex Korbonits]() Topics: [Automated reasoning](), [Large Language Model](), [Formal verification](), [Prompt Engineering](), [Claude](), [graph theory](), [Machine learning](), [Artificial Intelligence]() Tags: [automated-reasoning](), [claude](), [formal-verification](), [graph-theory](), [llm](), [prompt-engineering]() ### AI overview The author revisits a November 2023 idea about using Claude and prompt engineering to translate problems into propositions for an SMT solver, satisfy proofs, eliminate unknowns and quantifiers, and return results to an agent. The essay connects this idea to the author's background in formal logic, graph theory and machine learning, while noting that the concept was not novel in isolation. ### Source excerpt A November 2023 notebook entry about grounding LLMs with SMT solvers -- why formal verification might be the right fix for hallucination, and what it took to finally say it out loud. ## My Graduate Career in Math DevFeed: [My Graduate Career in Math]() Original publisher: [Read original article]() Published: 2025-05-12T18:35:57Z Content type: opinion Language: en Sources: [Jeremy Kun]() Topics: [Mathematics](), [Computer science](), [graph theory]() Tags: [computer-science](), [education](), [essay](), [essays](), [game-theory](), [graph-theory](), [group-theory](), [linear-algebra](), [math](), [mathematics](), [retrospective](), [topology](), [university]() ### AI overview An autobiographical essay about the author's transition from computer science to mathematics at Cal Poly, including university coursework, study abroad in Budapest, and reflections on the intellectual environment and an early group theory project. ### Source excerpt Editor's note: This essay was originally published on Medium on 2016-03-05. I have made minor edits in this republishing and added a few small retrospective notes. 2010-2011 (Year 0) I had just switched my major at Cal Poly State University from computer science to math. I wanted to double major but California was in a budget crisis and a few weeks before I tried submitting my double-major request the Provost for the CSU system put a blanket ban on double majors. ## Rendering the Java heap as a Treemap DevFeed: [Rendering the Java heap as a Treemap]() Original publisher: [Read original article]() Author: Pierre-Yves Ricau Published: 2024-09-25T04:37:22Z Content type: tutorial Language: en Sources: [Py's blog]() Topics: [Java](), [Graphs]() Tags: [android](), [computer-science](), [garbage-collection](), [gc](), [graph-theory](), [java](), [kotlin](), [memory](), [memory-management]() ### AI overview An exploration of Java heap dumps that explains shallow size, retained size, garbage-collection roots, and dominators. It also demonstrates using Shark and Graphviz to explore heap and view hierarchies. ### Source excerpt Exploring heap dumps I have investigated many heap dumps over the years, and I usually switch back and forth between two tools: YourKit Java Profiler to poke around the heap and look for interesting things (ask your company to buy a license!) Shark... ## Ideas and Solutions for Advent of Code 2021 in Kotlin -- Part 3/4 DevFeed: [Ideas and Solutions for Advent of Code 2021 in Kotlin -- Part 3/4]() Original publisher: [Read original article]() Author: Yev Kanivets Published: 2021-12-26T14:22:49Z Content type: tutorial Language: en Sources: [xorum.io - Medium]() Topics: [Advent of Code](), [Kotlin](), [Programming](), [competitive-programming](), [Graphs](), [math]() Tags: [advent-of-code](), [algorithms](), [array](), [code](), [competitive-programming](), [data-structures](), [graph-theory](), [kotlin](), [memory](), [pairs](), [programming](), [solutions](), [time]() ### AI overview This article presents high-level ideas and Kotlin solutions for Advent of Code 2021 tasks from days 13 to 15. It discusses folding a 2D array, modeling polymer growth with dynamic programming, and finding a shortest path in a 2D array. ### Source excerpt Ideas and Solutions for Advent of Code 2021 in Kotlin -- Part 3/4 The third week of Advent of Code requires more time and even some competitive programming knowledge like dynamic programming and graph theory. In this article, I share some high-level ideas and my solutions if you need a hint or a few to get that gold star. Ideas and Solutions for tasks 1 to 6 can be found here, for tasks 7 to 12 -- here. Day 13: Transparent Origami The 2D array contains two types of symbols -- . and #. This array can be folded horizontally and vertically multiple times. When folding # symbols replace . symbols, but not vice versa. Here is the complete task. This is a modeling task, which you can do on the 2D array itself, but the possible range is quite large, so you can hit a memory limit. The smarter solution would be to fold the # symbols (the initial input, actually). Such folding can be done with Kotlin's fold function using the initial value of # symbol positions and mapNotNull, which mirrors X or Y coordinates depending on the fold direction. Here is my solution. Day 14: Extended Polymerization We get the initial string (template) consisting of uppercase letters, which describe the initial state of the polymer. The list of pair insertion rules allows growing the initial polymer step by step exponentially. We need to model this growth during 10 (first sub-task) and 40 (second sub-task) steps. Here is the complete task. The first sub-task can be solved just by modeling, but even 20 steps are too long and require too much memory and time. The (much) better solution is based on dynamic programming. You can easily see that every pair grows independently of others (new elements are always added inside the pair). It means that the solution can be found for each pair separately and then combined. It doesn't speed up things sufficiently though. But makes the solution easier? Yes. Now, let's grow each pair step by step. So at each step, every pair (usually) produces two more pairs, which h ## Adventure Games and Eigenvalues DevFeed: [Adventure Games and Eigenvalues]() Original publisher: [Read original article]() Author: Evan Miller Published: 2017-04-23T10:10:00Z Content type: tutorial Language: en Sources: [Evan Miller]() Topics: [Mathematics](), [Algorithms, Complexity](), [Formal verification](), [Graphs]() Tags: [eigenvalues](), [formal-verification](), [graph-theory](), [matrix](), [state]() ### AI overview This article explains how Markov theory and matrix manipulation can prove the absence of dead ends in a small adventure game and quantify the risk of reaching invalid states. It compares this approach with formal verification and graph-based route-finding algorithms, while noting that all approaches become impractical for large state spaces. ### Source excerpt Identifying and quantifying dead ends, the bane of all adventure game players: Adventure Games and Eigenvalues ## When Greedy Algorithms are Perfect: the Matroid DevFeed: [When Greedy Algorithms are Perfect: the Matroid]() Original publisher: [Read original article]() Published: 2014-08-26T09:00:02Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Algorithms, Complexity](), [graph theory](), [Graphs](), [math]() Tags: [algorithms](), [graph-theory](), [graphs](), [greedy](), [greedy-algorithm](), [kruskal-s-algorithm](), [linear-independence](), [matroids](), [minimum-spanning-trees](), [trees](), [vertex]() ### AI overview This article explains when greedy algorithms are guaranteed to produce optimal solutions. It introduces matroids as the framework characterizing that guarantee and uses the minimum spanning tree problem as an example, with background on matroid history and connections to linear algebra and graph theory. ### Source excerpt Greedy algorithms are by far one of the easiest and most well-understood algorithmic techniques. There is a wealth of variations, but at its core the greedy algorithm optimizes something using the natural rule, "pick what looks best" at any step. So a greedy routing algorithm would say to a routing problem: "You want to visit all these locations with minimum travel time? Let's start by going to the closest one. And from there to the next closest one. ## Anti-Coordination Games and Stable Graph Colorings DevFeed: [Anti-Coordination Games and Stable Graph Colorings]() Original publisher: [Read original article]() Published: 2013-09-09T09:00:00Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [math]() Tags: [announce](), [anti-coordination](), [game-theory](), [games-on-graphs](), [graph](), [graph-coloring](), [graph-theory](), [network](), [np-completeness](), [paper](), [publication](), [research]() ### AI overview The author announces that a first paper on anti-coordination games and stable colorings has been accepted for publication. The article introduces the problem for a general audience, explaining how graph-based friendship networks affect interactions and raising questions about stability, convergence, centralized strategies, and computational difficulty. ### Source excerpt My First Paper I'm pleased to announce that my first paper, titled "Anti-Coordination Games and Stable Colorings," has been accepted for publication! The venue is the Symposium on Algorithmic Game Theory, which will take place in Aachen, Germany this October. A professor of mine once told me that everyone puts their first few publications on a pedestal, so I'll do my best to keep things down to earth by focusing on the contents of the paper and not my swirling cocktail of pride. ## The Erdős-Rényi Random Graph DevFeed: [The Erdős-Rényi Random Graph]() Original publisher: [Read original article]() Published: 2013-08-22T16:11:18Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [math](), [Networks]() Tags: [color](), [combinatorics](), [erdos](), [giant-component](), [graph](), [graph-coloring](), [graph-theory](), [properties](), [random-graph](), [random-graphs](), [renyi](), [social-networks](), [structure]() ### AI overview This tutorial introduces the Erdős-Rényi random graph model G(n,p), in which each possible edge between n vertices is included independently with probability p. It discusses the model's role in combinatorics and network analysis, along with basic properties, applications, and computational exploration. ### Source excerpt During the 1950's the famous mathematician Paul Erdős and Alfred Rényi put forth the concept of a random graph and in the subsequent years of study transformed the world of combinatorics. The random graph is the perfect example of a good mathematical definition: it's simple, has surprisingly intricate structure, and yields many applications. In this post we'll explore basic facts about random graphs, slowly detail a proof on their applications to graph theory, and explore their more interesting properties computationally (a prelude to proofs about their structure). ## Depth- and Breadth-First Search DevFeed: [Depth- and Breadth-First Search]() Original publisher: [Read original article]() Published: 2013-01-22T11:44:27Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [Algorithms](), [Data structures](), [graph theory](), [Programming](), [Python]() Tags: [algorithms](), [breadth-first-search](), [computer](), [computer-science](), [data-structures](), [depth-first-search](), [graph-theory](), [graphs](), [mathematics](), [optimization](), [programming](), [python](), [queue](), [stack]() ### AI overview This tutorial introduces graph search through depth-first search and breadth-first search. It reviews directed graphs, vertices, edges, adjacency functions, and implementations of the basic graph data structure in mathematical terms and Python. ### Source excerpt The graph is among the most common data structures in computer science, and it's unsurprising that a staggeringly large amount of time has been dedicated to developing algorithms on graphs. Indeed, many problems in areas ranging from sociology, linguistics, to chemistry and artificial intelligence can be translated into questions about graphs. It's no stretch to say that graphs are truly ubiquitous. Even more, common problems often concern the existence and optimality of paths from one vertex to another with certain properties. ## Neural Networks and the Backpropagation Algorithm DevFeed: [Neural Networks and the Backpropagation Algorithm]() Original publisher: [Read original article]() Published: 2012-12-09T21:24:48Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Neural Network](), [Algorithm](), [datasets](), [Graphs]() Tags: [algorithm](), [classes](), [data](), [gradient-descent](), [graph-theory](), [machine-learning](), [neural-networks](), [perceptron]() ### AI overview This tutorial introduces neurons as an extension of the perceptron model. It explains how multiple perceptron models can introduce nonlinearity into classification and begins defining a neuron using weighted inputs, an inner product, and a bias term. ### Source excerpt Neurons, as an Extension of the Perceptron Model In a previous post in this series we investigated the Perceptron model for determining whether some data was linearly separable. That is, given a data set where the points are labelled in one of two classes, we were interested in finding a hyperplane that separates the classes. In the case of points in the plane, this just reduced to finding lines which separated the points like this: ## Ramsey Number Lower Bound DevFeed: [Ramsey Number Lower Bound]() Original publisher: [Read original article]() Published: 2012-12-02T15:17:26Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [graph theory](), [Graphs](), [Mathematics]() Tags: [combinatorics](), [computational-complexity](), [graph](), [graph-coloring](), [graph-theory](), [mathematics](), [number-theory](), [probabilistic-method](), [ramsey-numbers](), [random-graph]() ### AI overview The article proves a lower bound for the diagonal Ramsey number R(m,m). By randomly coloring the edges of a complete graph, it shows that if binom(n,m)2^(1−binom(m,2)) is less than 1, there exists a coloring with no monochromatic m-vertex subgraph, so R(m,m) > n. ### Source excerpt Define the Ramsey number $ R(k,m)$ to be the minimum number $ n$ of vertices required of the complete graph $ K_n$ so that for any two-coloring (red, blue) of the edges of $ K_n$ one of two things will happen: There is a red $ k$-clique; that is, a complete subgraph of $ k$ vertices for which all edges are red. There is a blue $ m$-clique. It is known that these numbers are always finite, but it is very difficult to compute them exactly. ## Trees--A Primer DevFeed: [Trees--A Primer]() Original publisher: [Read original article]() Published: 2012-09-16T22:40:47Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [graph theory](), [Graphs](), [Algorithms](), [Learning]() Tags: [algorithms](), [graph](), [graph-theory](), [learning](), [mathematics](), [primer](), [programming](), [python](), [trees]() ### AI overview A primer on trees in graph theory. It defines trees as connected graphs without cycles, explains rooted-tree hierarchy, depth, leaves, and internal nodes, and introduces common algorithms on trees. ### Source excerpt This post comes in preparation for a post on decision trees (a specific type of tree used for classification in machine learning). While most mathematicians and programmers are familiar with trees, we have yet to discuss them on this blog. For completeness, we'll give a brief overview of the terminology and constructions associated with trees, and describe a few common algorithms on trees. We will assume the reader has read our first primer on graph theory, which is a light assumption. ## Machine Learning -- Introduction DevFeed: [Machine Learning -- Introduction]() Original publisher: [Read original article]() Published: 2012-08-04T19:59:42Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Machine Learning & Artificial Intelligence](), [Algorithms](), [data](), [Optimization](), [Mathematics](), [graph theory]() Tags: [algorithms](), [data](), [decision-trees](), [graph-theory](), [machine-learning](), [mathematics](), [neural-networks](), [optimization](), [support-vector-machines]() ### AI overview This introduction presents machine learning as a broad toolbox that draws on optimization, statistical inference, linear algebra, graph theory, functional analysis, and topology. It outlines a recurring process of developing algorithms or mathematical models, testing them on datasets, and refining them with domain knowledge, while emphasizing mathematical foundations, implementation efficiency, and the structure of natural data. ### Source excerpt A Series on Machine Learning These days an absolutely staggering amount of research and development work goes into the very coarsely defined field of "machine learning." Part of the reason why it's so coarsely defined is because it borrows techniques from so many different fields. Many problems in machine learning can be phrased in different but equivalent ways. While they are often purely optimization problems, such techniques can be expressed in terms of statistical inference, have biological interpretations, or have a distinctly geometric and topological flavor. ## Handshake Lemma DevFeed: [Handshake Lemma]() Original publisher: [Read original article]() Published: 2012-01-29T22:24:31Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [Mathematics]() Tags: [graph](), [graph-theory](), [handshake](), [mathematics]() ### AI overview The article proves the Handshake Lemma: in a party of n people, the number of people with an odd number of friends must be even. It uses double-counting to show that the sum of all friendship counts equals twice the total number of friendships, and then relates the result to graph theory. ### Source excerpt Problem: Prove or disprove: at a party of $ n$ people, there must be an even number of people who have an odd number of friends at the party. Solution: Let $ P$ be the set of all people, and for any person $ p \in P$, let $ d(p)$ be the number of friends that person has. Let $ f$ be the total number of friendships between pairs of people at the party. ## n-Colorability is Equivalent to Finite n-Colorability (A Formal Logic Proof) DevFeed: [n-Colorability is Equivalent to Finite n-Colorability (A Formal Logic Proof)]() Original publisher: [Read original article]() Published: 2011-09-04T22:41:27Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [Math and Logic]() Tags: [completeness](), [graph](), [graph-coloring](), [graph-theory](), [logic](), [vertex]() ### AI overview A formal logic proof shows that an infinite graph is n-colorable exactly when every finite subgraph is n-colorable. The proof encodes valid colorings as propositional formulas and applies the Compactness Theorem. ### Source excerpt Warning: this proof requires a bit of familiarity with the terminology of propositional logic and graph theory. Problem: Let $ G$ be an infinite graph. Show that $ G$ is $ n$-colorable if and only if every finite subgraph $ G_0 \subset G$ is $ n$-colorable. Solution: One of the many equivalent versions of the Compactness Theorem for the propositional calculus states that if $ \Sigma \subset \textup{Prop}(A)$, where $ A$ is a set of propositional atoms, then $ \Sigma$ is satisfiable if and only if any finite subset of $ \Sigma$ is satisfiable. ## Graph Coloring, or Proof by Crayon DevFeed: [Graph Coloring, or Proof by Crayon]() Original publisher: [Read original article]() Published: 2011-07-14T22:24:13Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [color](), [math]() Tags: [color](), [graph](), [graph-coloring](), [graph-theory](), [graphs](), [map](), [maps](), [mathematics](), [primer](), [pseudocode]() ### AI overview This tutorial introduces graph coloring through map-coloring examples, including the provinces of Costa Rica and the arrondissements of Paris. It models adjacent regions as vertices connected by edges, defines a valid graph coloring, and introduces the chromatic number as the smallest number of colors needed. ### Source excerpt How many colors are required to color the provinces of Costa Rica? A common visual aid for maps is to color the regions of the map differently, so that no two regions which share a border also share a color. For example, to the right is a map of the provinces of Costa Rica (where the author is presently spending his vacation). It is colored with eight different colors, one for each province. ## Teaching Mathematics--Graph Theory DevFeed: [Teaching Mathematics--Graph Theory]() Original publisher: [Read original article]() Published: 2011-06-26T16:19:20Z Content type: tutorial Language: en Sources: [Jeremy Kun]() Topics: [graph theory](), [Graphs](), [Mathematics]() Tags: [campolindo](), [education](), [google](), [graph-theory](), [introspection](), [mathematics](), [methods-of-proof](), [primer](), [puzzles](), [seven-bridges](), [seven-hills](), [teaching]() ### AI overview A mathematics teacher describes using puzzles and student-led discovery to introduce graph theory in high-school and middle-school lectures. The article presents the Seven Bridges of Königsberg problem as a central example and discusses mathematics as a non-linear process of questions, conjectures, and proof. ### Source excerpt Community Service Mathematics is supposed to be a process of discovery. Definitions, propositions, and methods of proof don't come from nowhere, although after the fact (when presented in a textbook) they often seem to. As opposed to a textbook, real maths is highly non-linear. It took mathematicians quite a lot of fuss to come up with the quadratic formula, and even simple geometric conjectures were for the longest time the subject of hot debate. ## Google's PageRank--A First Attempt DevFeed: [Google's PageRank--A First Attempt]() Original publisher: [Read original article]() Published: 2011-06-18T18:05:20Z Content type: article Language: en Sources: [Jeremy Kun]() Topics: [Graphs](), [graph theory](), [Web](), [Algorithm](), [Internet](), [structure]() Tags: [algorithm](), [analysis](), [eigenvalues](), [eigenvectors](), [google](), [graph](), [graph-theory](), [internet](), [mathematics](), [page-rank](), [ranking](), [search-engine](), [structure](), [vertex](), [web]() ### AI overview This post models the Web as a directed graph and introduces PageRank-style importance scoring for web pages. It first considers ranking pages by incoming-link counts, then explains why treating every link as equally valuable is inadequate and motivates weighting links by the importance of the linking page. ### Source excerpt The Web as a Graph The goal of this post is to assign an "importance score" $ x_i \in [0,1]$ to each of a set of web pages indexed $ v_i$ in a way that consistently captures our idea of which websites are likely to be important. But before we can extract information from the structure of the internet, we need to have a mathematical description of that structure. Enter graph theory.