Abstract. An upper bound for the number of Hamiltonian cycles of symmetric diagraphs is established first in this paper, which is tighter than the famous Minc’s bound and the Brégman’s bound. A transformation on graphs is proposed, so that counting the number of Hamiltonian cycles of an undirected graph can be done by counting the number of Hamiltonian cycles of its corresponding symmetric dire...

We prove that random d-regular Cayley graphs of the symmetric group asymptotically almost surely have girth at least (logd−1 |G|) /2 and that random d-regular Cayley graphs of simple algebraic groups over Fq asymptotically almost surely have girth at least logd−1 |G|/ dim(G). For the symmetric p-groups the girth is between log log |G| and (log |G|) with α < 1. Several conjectures and open quest...

In this paper we apply Pólya’s Theorem to the problem of enumerating Cayley graphs on permutation groups up to isomorphisms induced by conjugacy in the symmetric group. We report the results of a search of all three-regular Cayley graphs on permutation groups of degree at most nine for small diameter graphs. We explore several methods of constructing covering graphs of these Cayley graphs. Exam...

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variab...

The chromatic symmetric function of a graph is a symmetric function that generalizes the chromatic polynomial. Its investigation has largely been motivated by the existence of an open problem, the poset-chain conjecture, which is equivalent to the assertion that for certain graphs, the coefficients in the expansion of the chromatic symmetric function in terms of elementary symmetric functions, ...

Symmetry is a widespread phenomenon that can offer opportunities for powerful exploitation in areas as diverse as molecular chemistry, pure mathematics, circuit design, biology and architecture. Graphs are an abstract way to represent relational structures. The search for symmetry in many contexts can thus be reduced to the attempt to find graph automorphisms. Brendan McKay’s NAUTY system (McKa...

Domination is a well-known graph theoretic concept due to its significant real-world applications in several domains, such as design and communication network analysis, coding theory, optimization. For connected ? = V , E ...