On basic graphs of symmetric graphs of valency five

TL;DR

This paper classifies symmetric basic graphs of order 2qp^n and valency 5, revealing their structure as Cayley graphs on dihedral groups, complete graphs, bipartite graphs, or sporadic coset graphs, with applications to small order graphs.

Contribution

It provides a complete classification of symmetric basic graphs of order 2qp^n and valency 5, including new families and sporadic examples, extending previous classifications.

Findings

Graphs are isomorphic to Cayley graphs on dihedral groups with specific divisibility conditions.

Includes the complete graph K_6 and bipartite graph K_{5,5} as special cases.

Identifies nine sporadic coset graphs related to non-abelian simple groups.

Abstract

A graph $\G$ is {\em symmetric} or {\em arc-transitive} if its automorphism group $\Aut(\G)$ is transitive on the arc set of the graph, and $\G$ is {\em basic} if $\Aut(\G)$ has no non-trivial normal subgroup $N$ such that the quotient graph $\G_N$ has the same valency with $\G$. In this paper, we classify symmetric basic graphs of order $2qp^n$ and valency 5, where $q<p$ are two primes and $n$ is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order $2q$ with $5\di (q-1)$, the complete graph $K_6$ of order $6$, the complete bipartite graph $K_{5,5}$ of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order $kp^n$ for some small integers $k$ and $n$ are classified.