twelvevertex

Twelvevertex is a term used in graph theory to denote graphs with exactly twelve vertices. In practice, it may refer to the collection of all simple graphs on twelve vertices or to discussions of specific twelve-vertex graphs in enumeration, classification, and network design.

There are 2^66 simple graphs on twelve vertices, since there are 66 possible edges between pairs of vertices. Researchers classify twelvevertex graphs by properties such as connectedness, regularity, and degree sequence. Common families include the cycle C12, the path P12, the complete graph K12, and the complete bipartite graph K6,6; among twelve-vertex graphs, several special structures are studied for their symmetry or planarity. A notable example is the icosahedral graph, which has 12 vertices, 30 edges, is 5-regular, planar, and vertex-transitive.

Core properties considered for twelvevertex graphs include degree sequence, chromatic number, automorphism group, spectrum (eigenvalues of the adjacency matrix), and connectivity. Because the vertex count is small enough to enumerate exhaustively, twelvevertex graphs serve as test cases in graph-isomorphism algorithms and in explorations of extremal graph theory.

Applications of twelvevertex graphs appear in theoretical studies, teaching, and computational experiments in network topology and coding theory.

See also: isomorphism of graphs, graph theory, icosahedral graph.