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An automorphism of a graph is a graph isomorphism with itself, i.e., a mapping from the vertices of the given graph G back to vertices of G such that the resulting graph is ...

A cactus graph, sometimes also called a cactus tree, a mixed Husimi tree, or a polygonal cactus with bridges, is a connected graph in which any two graph cycles have no edge ...

Vizing's theorem states that a graph can be edge-colored in either Delta or Delta+1 colors, where Delta is the maximum vertex degree of the graph. A graph with edge chromatic ...

A graph G is distance transitive if its automorphism group is transitive on pairs of vertices at each pairwise distance in the graph. Distance-transitivity is a ...

The Paley graph of order q with q a prime power is a graph on q nodes with two nodes adjacent if their difference is a square in the finite field GF(q). This graph is ...

The alternating group graph AG_n is the undirected Cayley graph of the set of 2(n-2) generators of the alternating group A_n given by g_3^-, g_3^+, g_4^-, g_4^+, ..., g_n^-, ...

The Harary graph H_(k,n) is a particular example of a k-connected graph with n graph vertices having the smallest possible number of edges. The smallest number of edges ...

A graph G is hypohamiltonian if G is nonhamiltonian, but G-v is Hamiltonian for every v in V (Bondy and Murty 1976, p. 61). The Petersen graph, which has ten nodes, is the ...

The utility problem posits three houses and three utility companies--say, gas, electric, and water--and asks if each utility can be connected to each house without having any ...

The edge set of a graph is simply a set of all edges of the graph. The cardinality of the edge set for a given graph g is known as the edge count of g. The edge set for a ...