Graphtheoretical

Graph theory, the graphtheoretical study of graphs, is a branch of mathematics and computer science that analyzes graphs—structures consisting of vertices (nodes) connected by edges. Graphs can be undirected or directed (digraphs), weighted or unweighted, and simple or multigraphs. They are used to model pairwise relations in networks such as transportation, communication, social, and biological systems.

Core concepts include connectivity, paths and cycles, trees (connected acyclic graphs), degree (the number of incident edges), adjacency, and planarity (graphs that can be drawn without crossing edges). Important structures include matchings, coverings, and colorings; common problems involve finding shortest paths, constructing minimum spanning trees, and determining network flows.

The algorithmic toolkit encompasses breadth-first and depth-first search for traversal; Dijkstra’s algorithm for shortest paths with nonnegative weights; Bellman-Ford for negative weights; Kruskal’s and Prim’s algorithms for minimum spanning trees; and Ford–Fulkerson and related methods for maximum-flow problems. Many graph problems are computationally challenging: several decision and optimization questions on general graphs are NP-hard, such as Hamiltonian path and graph coloring, while others have efficient solutions for restricted graph classes. The graph isomorphism problem concerns determining whether two graphs are structurally identical under relabeling.

Applications span computer networks, routing and logistics, social-network analysis, chemistry and bioinformatics, circuit design, and scheduling. The field has a long history dating to Euler’s Königsberg bridges problem and has grown through contributions from numerous mathematicians and computer scientists.