# Symmetric graphs of valency seven and their basic normal quotient graphs

## TL;DR

This paper classifies basic normal quotient graphs of all connected 7-valent symmetric graphs with specific order structures, revealing finiteness results and generalizing previous theorems in graph symmetry.

## Contribution

It completely determines the basic normal quotient graphs of certain 7-valent symmetric graphs of order 2pq^n, including an infinite family and specific small graphs, extending prior work.

## Key findings

- Identifies an infinite family of dihedrants of order 2p with p ≡ 1 (mod 7)
- Classifies 6 specific graphs with order up to 310
- Shows finiteness of 2-arc-transitive 7-valent graphs of certain orders

## Abstract

A graph $\Gamma$ is basic if Aut$\Gamma$ has no normal subgroup $N\ne1$ such that $\Gamma$ is a normal cover of the normal quotient graph $\Gamma_N$. In this paper, we completely determine the basic normal quotient graphs of all connected 7-valent symmetric graphs of order $2pq^n$ with $p < q$ odd primes, which consist of an infinite family of dihedrants of order $2p$ with $p\equiv1$(mod 7), and 6 specific graphs with order at most 310. As a consequence, it shows that, for any given positive integer n, there are only finitely many connected 2-arc-transitive 7-valent graphs of order $2pq^n$ with $7\ne p<q$ primes, partially generalizing Theorem 1 of Conder, Li and Potocnik [On the orders of arc-transitive graphs, J. Algebra 421 (2015), 167-186].