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A graph models relationships using vertices and edges; edges may be directed or undirected and may have weights.
Graph theory is a branch of discrete mathematics that studies graphs, mathematical structures used to model pairwise relationships between objects. A graph consists of vertices, also called nodes, connected by edges. Graphs may be undirected, directed, mixed, simple, or multigraphs, and edges may carry weights. Important concepts include endpoints, loops, paths, connectivity, graph coloring, graph embeddings, and spanning trees. The field originated with Leonhard Euler's 1736 analysis of the Seven Bridges of Königsberg and has since developed into several major subareas. Topological graph theory examines embeddings, planarity, crossings, and graph minors; algebraic graph theory studies matrices, eigenvalues, and symmetry; geometric graph theory considers spatial representations and intersections; extremal graph theory investigates maximum edge counts under restrictions; and random graph theory uses probability to study graph structures and models. Graph theory has broad applications in computer science, network analysis, chemistry, engineering, biology, and social sciences.
A graph models relationships using vertices and edges; edges may be directed or undirected and may have weights.
Major graph types include simple graphs, multigraphs, mixed graphs, weighted graphs, trees, planar graphs, and random graphs.
Graph theory includes subareas such as topological, algebraic, geometric, extremal, and random graph theory.
Foundational results include Euler's work on the Seven Bridges of Königsberg, the four-color theorem, Mantel's and Turán's theorems, and the theory of graph minors.
Graphs are widely applied to networks, routing, scheduling, circuit analysis, data visualization, biological systems, and social relationships.
A vertex is a fundamental point or node in a graph that represents an object.
An edge is a connection between two vertices and represents a relationship between them.
A directed graph is a graph whose edges have an assigned direction represented by arrows.
An undirected graph is a graph whose edges represent symmetric relationships without directions.
A simple graph has no multiple edges between the same pair of vertices and typically has no loops.
A multigraph allows multiple edges between the same pair of vertices and may allow loops.
A tree is a connected undirected graph in which every pair of vertices is connected by exactly one path.
A planar graph can be drawn on a plane without any two edges crossing except at their endpoints.
Graph coloring assigns labels called colors to graph elements so that adjacent elements receive different colors.
A graph minor is obtained from a graph by deleting vertices or edges and contracting edges.
An adjacency matrix is a matrix that records which pairs of vertices in a graph are connected by edges.
A graph invariant is a property that depends on a graph's structure rather than on its labels or drawing.
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