Хьювуд графигі: 14 төбесі бар кубтық график

Undirected graph with 14 vertices

In the mathematical field of graph theory, the Heawood graph is an undirected graph with 14 vertices and 21 edges, named after Percy John Heawood.

The graph is cubic, and all cycles in the graph have six or more edges. Every smaller cubic graph has shorter cycles, so this graph is the 6 cage, the smallest cubic graph of girth 6. It is a distance transitive graph (see the Foster census) and therefore distance regular. There are 24 perfect matchings in the Heawood graph; for each matching, the set of edges not in the matching forms a Hamiltonian cycle. For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, 3 color its edges) in eight different ways. There are 28 six vertex cycles in the Heawood graph. Each 6 cycle is disjoint from exactly three other 6 cycles; among these three 6 cycles, each one is the symmetric difference of the other two. The graph with one node per 6 cycle, and one edge for each disjoint pair of 6 cycles, is the Coxeter graph.

The automorphism group of the Heawood graph is isomorphic to the projective linear group PGL2(7), a group of order 336. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore, the Heawood graph is a symmetric graph. It has automorphisms that take any vertex to any other vertex and any edge to any other edge. More strongly, the Heawood graph is 4 arc transitive. According to the Foster census, the Heawood graph, referenced as F014A, is the only cubic symmetric graph on 14 vertices. It has book thickness 3 and queue number 2. The characteristic polynomial of the Heawood graph is It is the only graph with this characteristic polynomial, making it a graph determined by its spectrum.