The descendant set desc(α) of a vertex α in a digraph D is the set of vertices which can be reached by a directed path from α. A subdigraph of D is finitely generated if it is the union of finitely many descendant sets and D is descendant-homogeneous if it is vertex transitive and any isomorphism between finitely generated subdigraphs extends to an automorphism. We consider connected descendant...

A graph is symmetric if its automorphism group acts transitively on the set of arcs of the graph. In this paper, we classify connected heptavalent symmetric graphs of order 6p for each prime p. As a result, there are three sporadic such graphs: one for p = 5 and two for p = 13.

symmetric division deg index is one of the 148 discrete adriatic indices that showed good predictive properties on the testing sets provided by international academy of mathematical chemistry.symmetric division deg index is defined by$$sdd(g) = sume left( frac{min{d_u,d_v}}{max{d_u,d_v}} + frac{max{d_u,d_v}}{min{d_u,d_v}} right),$$where $d_i$ is the degree of vertex $i$ in graph $g$.in this pap...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set and edge set, but not arc set. It was shown by [Y.-Q. Feng, K.S. Wang, C.X. Zhou, Tetravalent halftransitive graphs of order 4p, European J. Combin. 28 (2007) 726–733] that all tetravalent half-arc-transitive graphs of order 4p for a prime p are non-Cayley and such graphs exist if andonly if p−1 is divi...

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime inor out-degree. Two differe...

Let p be a prime and n a positive integer. In [J. Austral. Math. Soc. 81 (2006), 153–164], Feng and Kwak showed that if p > 5 then every connected cubic symmetric graph of order 2p is a Cayley graph. Clearly, this is not true for p = 5 because the Petersen graph is non-Cayley. But they conjectured that this is true for p = 3. This conjecture is confirmed in this paper. Also, for the case when p...

Graph homomorphism has been studied intensively. Given an m ×m symmetric matrix A, the graph homomorphism function is defined as

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p2 exist and a tetravalent half-arctransitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p − 1 is divisible by 8 and it is unique for a given o...

An arc of a graph is an oriented edge and a 3-arc is a 4-tuple (v, u, x, y) of vertices such that both (v, u, x) and (u, x, y) are paths of length two. The 3-arc graph of a given graph G, X(G), is defined to have vertices the arcs of G. Two arcs uv, xy are adjacent in X(G) if and only if (v, u, x, y) is a 3-arc of G. This notion was introduced in recent studies of arc-transitive graphs. In this...

We consider two definitions of the even-dimensional hypercube given in the literature. The labeled graphs obtained by two definitions are not same, but one is isomorphic to the other. By interconnecting two labeled graphs in such a way that each pair of vertices with the same label are joined by an edge, we construct a vertex-symmetric graph with the diameter about half that of a comparable hyp...