--- title: Graph Theory description: Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Learners will understand how to analyze network connectivity, find shortest paths, and solve optimization problems. category: mathematics subcategory: combinatorics difficulty: beginner, intermediate, advanced url: /subject/graph-theory --- # Graph Theory Graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. Learners will understand how to analyze network connectivity, find shortest paths, and solve optimization problems. ## Available Resources 2 Books • 4 Courses • 5 Websites ## Websites ### 1. House of Graphs Searchable repository from Ghent University holding graphs that appear as extremal examples or counterexamples, each with precomputed invariants such as chromatic number, diameter, and girth. Use it to test conjectures and find concrete examples quickly. **Difficulty:** Beginner | **Price:** Free **Link:** https://houseofgraphs.org/ **Tags:** graph-theory, graph-database, graph-invariants, combinatorics, reference ### 2. Diestel Graph Theory Official site for Reinhard Diestel's Springer graduate text, where the main text is readable free online and paid eBook editions add the full apparatus. It covers matching, connectivity, planarity, colouring, flows, extremal theory, and minors. **Difficulty:** Beginner | **Price:** Free **Link:** https://diestel-graph-theory.com/ **Tags:** graph-theory, textbook, discrete-mathematics, combinatorics, free-online ### 3. Visualgo.net Interactive visualization module from VisuAlgo, built at the National University of Singapore, showing how graphs are stored as adjacency matrices, adjacency lists and edge lists, so learners can compare these representations and trace graph operations step by step. **Difficulty:** Beginner | **Price:** Free **Link:** https://visualgo.net/en/graphds **Tags:** graph-representation, adjacency-list, adjacency-matrix, data-structures, algorithm-visualization ### 4. GeeksforGeeks Graphs Tutorials and problems **Difficulty:** Beginner | **Price:** Free **Link:** https://www.geeksforgeeks.org/graph-data-structure-and-algorithms/ ### 5. Wolfram MathWorld **Author:** Eric W. Weisstein MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://mathworld.wolfram.com **Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry ## Courses ### 1. Graph Theory Learn graph theory basics and advanced concepts with Trefor Bazett's engaging course! Explore networks, algorithms, and more. **Difficulty:** Advanced | **Price:** Free **Link:** https://www.youtube.com/playlist?list=PLt5AfwLFPxWIO0rkWl44MhS_PLLqu3Kvr **Tags:** graph-theory, discrete-mathematics, video-lectures, combinatorics ### 2. Mathematics for Computer Science (MIT 6.042J) **Author:** Tom Leighton, Marten van Dijk Discrete mathematics for computer science with an emphasis on definitions and proofs: logic, induction, sets and relations, graph theory, modular arithmetic, asymptotics, counting and discrete probability. 25 lecture videos, problem sets and exams with solutions build fluency in writing proofs. **Difficulty:** Intermediate | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/ **Tags:** discrete-mathematics, mathematical-proofs, induction, graph-theory, combinatorics, discrete-probability ### 3. Discrete Mathematics **Author:** Dominik Scheder Dominik Scheder's proof-based course on sets, functions and relations, enumerative combinatorics, graph theory, and network flows and matchings, each concept paired with a fully proved non-trivial result. Learners read formal statements and write rigorous proofs of their own. **Difficulty:** Intermediate | **Language:** English | **Duration:** 11 weeks of study, 3-5 hours per week. | **Price:** Free **Link:** https://www.coursera.org/learn/discrete-mathematics **Tags:** discrete-math, combinatorics, graph-theory, network-flows, proofs ### 4. Introduction to Graph Theory We invite you to a fascinating journey into Graph Theory — an area which connects the elegance of painting and the rigor of mathematics; is simple, but not unsophisticated. Graph Theory gives us, both an easy way to pictorially represent many major mathematical results, and insights into the deep theories behind them. In this online course, among other intriguing applications, we will see how GPS systems find shortest routes, how engineers design integrated circuits, how biologists assemble genomes, why a political map can always be colored using a few colors. We will study Ramsey Theory which proves that in a large system, complete disorder is impossible! By the end of the course, we will implement an algorithm which finds an optimal assignment of students to schools. This algorithm, developed by David Gale and Lloyd S. Shapley, was later recognized by the conferral of Nobel Prize in Economics. As prerequisites we assume only basic math (e.g., we expect you to know what is a square or how to add fractions), basic programming in python (functions, loops, recursion), common sense and curiosity. Our intended audience are all people that work or plan to work in IT, starting from motivated high school students. **Difficulty:** Beginner | **Language:** English | **Duration:** 5 weeks, 3-5 hours/week | **Price:** Free **Link:** https://www.coursera.org/learn/graphs **Tags:** courses, mathematics-statistics, combinatorics ## Books ### 1. Introduction to Graph Theory **Author:** Douglas B. West Douglas West's standard graduate textbook covers trees, matchings, connectivity, colorings, planarity, and extremal problems, with proofs stated in full and extensive exercise sets. Working through it gives you the vocabulary and proof techniques used across combinatorics research. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/8120321421?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, combinatorics ### 2. Modern Graph Theory **Author:** Béla Bollobás Bollobás presents graph theory from an advanced standpoint, connecting extremal and random graphs to algebraic methods, electrical networks, and the Tutte polynomial. Readers finish able to follow modern research papers and apply probabilistic arguments to combinatorial problems. **Difficulty:** Intermediate | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/0387984887?tag=edmonddante07-20 **Tags:** books, mathematics-statistics, combinatorics --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/graph-theory