Pentavalent symmetric graphs of order 2p 3

Abstract

A graph is said to be symmetric if its automorphism group acts transitively on its arcs. In this paper, a complete classification of connected pentavalent symmetric graphs of order 12p is given for each prime p. As a result, a connected pentavalent symmetric graph of order 12p exists if and only if p = 2, 3, 5 or 11, and up to isomorphism, there are only nine such graphs: one for each p = 2, 3 and 5, and six for p = 11.

Heptavalent Symmetric Graphs of Order 16p

“…Guo [10] determined the exact structure of pentavalent case. Following this structure, a series of pentavalent symmetric graphs was classified in [15,19,20,26,27,12]. Recently, Guo [11] gave the exact structure of heptavalent case.…”

Arc-transitive cyclic and dihedral covers of pentavalent symmetric graphs of order twice a prime

“…Arc-transitive covers of non-simple graphs were also considered in literature. For example, regular covers of the dipole Dip k (a graph with two vertices and k parallel edges) were extensively studied in [1,15,25,26,32]. Such covers are called Haar graphs, and in particular, cyclic regular covers of dipoles are called cyclic Haar graphs, which can be regarded as a generalization of bipartite circulants and were studied in [20] (also see [14]).…”

“…Construction of Haar graphs have aroused wide concern. Marušič et al [25] studied elementary abelian covers of the dipole Dip p for a prime p. In particular, symmetric elementary abelian covers and Z 2 p × Z p -covers for a prime p of the dipole Dip 5 were classified completely in [15] and [32], respectively. Let p be a prime.…”

“…Pentavalent symmetric graphs of order 2p were classified by Cheng and Oxley in [5], which are the complete graph K 6 of order 6 and a family of Cayley graphs CD p with p = 5 or 5 (p − 1) on dihedral groups (see Proposition 3.4). It has been shown that many pentavalent symmetric graphs are regular covers of them, see [15,32]. In this paper, we consider arc-transitive cyclic and dihedral covers of these graphs.…”

“…Note that vertex-transitive bi-Cayley graphs of valency 3 over abelian groups were determined in [35], while the case for valency 5 is still elusive. Indeed, even for arctransitive pentavalent bi-Cayley graphs over abelian groups, it seems to be very difficult to give a classification, and one may see [1,15,32] for partial works.…”