Tetravalent Vertex- and Edge-Transitive Graphs Over Doubled Cycles

TL;DR

This paper classifies certain 4-valent symmetric graphs using algebraic coding theory and automorphism lifting, extending previous results and providing a unified framework for known cases.

Contribution

It introduces a novel approach linking graph symmetry properties to cyclic and negacyclic codes via automorphism lifting, generalizing earlier classifications.

Findings

Classification of 4-valent vertex- and edge-transitive graphs over doubled cycles.

Connection between graph symmetries and properties of generating polynomials of codes.

Unified description of previously resolved and unresolved cases.

Abstract

In order to complete (and generalize) results of Gardiner and Praeger on 4-valent symmetric graphs (European J. Combin, 15 (1994)) we apply the method of lifting automorphisms in the context of elementary-abelian covering projections. In particular, the vertex- and edge-transitive graphs whose quotient by a normal $p$-elementary abelian group of automorphisms, for $p$ an odd prime, is a cycle, are described in terms of cyclic and negacyclic codes. Specifically, the symmetry properties of such graphs are derived from certain properties of the generating polynomials of cyclic and negacyclic codes, that is, from divisors of $x^n \pm 1 \in {\mathbb Z}_p[x]$. As an application, a short and unified description of resolved and unresolved cases of Gardiner and Praeger are given.