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🔗 Graph Theory

Graph Algorithms • Network Analysis • Trees • Paths • Connectivity • Graph Representations

Master the mathematics of connections — from fundamental graph concepts and traversal algorithms to network analysis, graph coloring, and applications in computer science.

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Graph Theory Concepts & Algorithms

Graph Theory Basics

Vertices, edges, degree, paths, cycles.

Graph Representations

Adjacency matrix, adjacency list, incidence.

Data Structures

Graph Terminology

Degree, bipartite, complete, regular, planar.

BFS (Breadth-First Search)

Queue, level order, shortest path unweighted.

DFS (Depth-First Search)

Stack, recursion, topological sort, connectivity.

Topological Sort

DAG, ordering, Kahn's algorithm.

Dijkstra's Algorithm

Shortest path, priority queue, greedy.

Bellman-Ford

Negative weights, relaxation, cycles.

Floyd-Warshall

All-pairs shortest path, DP, transitive closure.

Johnson's Algorithm

All-pairs, reweighting, Dijkstra on all.

Kruskal's Algorithm

MST, union-find, sorting edges.

Prim's Algorithm

MST, priority queue, cut property.

Borůvka's Algorithm

MST, parallel, component-based.

Max Flow / Min Cut

Ford-Fulkerson, Edmonds-Karp, Dinic.

Dinic's Algorithm

Level graph, blocking flow, efficient.

Push-Relabel Algorithm

Preflow, relabel, height labels.

Graph Coloring

Chromatic number, greedy, four-color theorem.

Edge Coloring

Vizing's theorem, chromatic index.

Trees

Properties, rooted, binary, spanning trees.

Bipartite Graphs

Properties, matching, vertex cover.

Planar Graphs

Euler's formula, planar embeddings, dual.

Eulerian Graphs

Euler circuits, trails, Fleury's algorithm.

Hamiltonian Graphs

Hamiltonian cycles, paths, NP-complete.

Graph Connectivity

Connected components, bridges, articulation points.

Strongly Connected Components

Kosaraju, Tarjan, condensation graph.

Biconnected Components

Articulation points, bridges, DFS.

Matching Theory

Perfect matching, augmenting paths, König.

Hungarian Algorithm

Assignment problem, bipartite matching.

A* Search Algorithm

Heuristic, optimal, pathfinding.

Traveling Salesman Problem

NP-hard, approximations, heuristics.

Christofides Algorithm

TSP approximation, MST, perfect matching.

Graphs in Networking

Routing, network topology, graph algorithms.

Social Network Analysis

Centrality, PageRank, community detection.

Graph Databases

Neo4j, Gremlin, Cypher, graph queries.

Graph Neural Networks

GCN, GAT, graph embeddings.

Random Graphs

Erdos-Renyi, Gilbert, properties.

Small-World Networks

Watts-Strogatz, clustering, diameter.

Graph Isomorphism

Isomorphic graphs, Weisfeiler-Lehman.

Spectral Graph Theory

Eigenvalues, Laplacian, adjacency matrix.

Graph Algorithm Complexity

Time/space analysis, NP-hard, approximations.

Graph Libraries

NetworkX, GraphX, igraph, JGraphT.

NetworkX Analysis

Python library, graph algorithms, visualization.

Dynamic Graph Algorithms

Incremental, decremental, fully dynamic.

Topological Indices

Wiener, Randić, Zagreb, molecular graphs.

🔗 Graph Theory Mastery Roadmap

🔰 Foundation

Basic concepts, representations, BFS, DFS

📶 Core

Shortest paths, MST, trees, bipartite, coloring

🔥 Deep Dive

Network flow, matching, planar, Euler/Hamiltonian

🏆 Frontier

GNN, spectral theory, dynamic graphs, random graphs

🔗 Connect the Dots — Master Graph Theory

Interactive algorithms, network visualizations, and real-world graph applications.

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