grafiteoriaan

Grafiteoriaan (graph theory) is a branch of mathematics that studies graphs, abstract structures used to model pairwise relations between objects. A graph G consists of a set V of vertices and a set E of edges, where each edge connects two vertices. Graphs can be undirected or directed, and may be simple or may include multiple edges and loops; weighted graphs assign a numerical value to edges. Graph theory provides a language for representing networks, schedules, and structures across disciplines.

The field traces its origins to Euler’s solution of the Königsberg bridge problem in 1736, which established the idea of studying the existence and structure of paths. The term graph theory emerged in the 19th century as mathematicians explored graph-like representations of problems. Since then, graph theory has grown into a central area of combinatorics with connections to algebra, topology, probability, and computer science.

Key concepts include paths, cycles, connectivity, degree, and subgraphs. Important special classes are trees (connected acyclic graphs), bipartite graphs, and planar graphs. Fundamental problems concern Eulerian and Hamiltonian paths, graph coloring, matchings, and network flows. Major algorithmic results include breadth-first search and depth-first search; Dijkstra’s algorithm for shortest paths; Kruskal’s and Prim’s algorithms for minimum spanning trees; and the max-flow min-cut theorem.

Applications are broad: computer networks and routing, circuit design, social network analysis, chemistry and biology, and optimization problems in logistics and scheduling. Ongoing research areas include spectral graph theory, random graphs, graph algorithms, and, more recently, graph neural networks that learn from graph-structured data.