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graph theory
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A diagram of the historic Seven Bridges of Konigsberg, showing seven numbered bridges crossing over a river with multiple land masses, representing the classic mathematical problem. Five diagrams illustrate types of graphs in mathematics: a simple graph, a multigraph with parallel edges, a complete graph with all possible connections, a graph with a loop, and a directed graph with arrows showing direction. A geometric diagram illustrating an Eulerian circuit, with arrows tracing a continuous path along intersecting straight lines that form a closed shape connecting five points. Hamiltonian circuit A diagram of the complete graph K5, showing five points with every pair connected by straight lines, forming a dense network where several edges cross each other, illustrating its non-planar nature. Two geometric diagrams compare the complete graphs K4 and K5, showing that K4 can be drawn without edges crossing (planar), while K5 cannot be drawn this way (nonplanar). A geometric diagram of the complete bipartite graph K3,2, showing five vertices with three on one side and two on the other, all connected by curved lines representing edges between each pair of opposite sets. Illustration of the gas, water, and electricity problem showing three utility buildings labeled W, G, and E, and three houses labeled A, B, and C. Colored lines connect each utility to each house, forming a complete bipartite graph K3,3, with a key identifying water pipes, gas lines, and electrical wires. The diagram demonstrates the non-planar nature of K3,3 in graph theory, as the connections cross over each other.
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A diagram of the historic Seven Bridges of Konigsberg, showing seven numbered bridges crossing over a river with multiple land masses, representing the classic mathematical problem.
bridges of Königsberg In the 18th century the Swiss mathematician Leonhard Euler was intrigued by the question of whether a route existed that would traverse each of the seven bridges exactly once. In demonstrating that the answer is no, he laid the foundation for graph theory.

graph theory

mathematics
Written by
Stephan C. Carlson
Professor of Mathematics, Rose-Hulman Institute of Technology, Terre Haute, Indiana. Author of Topology of Surfaces, Knots, and Manifolds: A First Undergraduate Course.
Stephan C. Carlson
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Encyclopaedia Britannica's editors oversee subject areas in which they have extensive knowledge, whether from years of experience gained by working on that content or via study for an advanced degree....
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graph theory, branch of mathematics concerned with networks of points connected by lines. The subject of graph theory had its beginnings in recreational math problems (see number game), but it has grown into a significant area of mathematical research, with applications in chemistry, operations research, social sciences, and computer science. The history of graph theory may be specifically traced to 1735, when the Swiss mathematician Leonhard Euler solved the Königsberg bridge problem. The Königsberg bridge...

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