Cores of imprimitive symmetric graphs of order a product of two distinct primes

TL;DR

This paper investigates the cores of imprimitive symmetric graphs with order as a product of two distinct primes, providing complete classifications in many cases and conditions for when the core is trivial or isomorphic to specific graphs.

Contribution

It offers a comprehensive analysis of the cores of such graphs, including complete classifications and conditions for their structure, advancing understanding of their automorphism properties.

Findings

The core of many imprimitive symmetric graphs is explicitly determined.

Conditions are established for when the graph itself is a core or its core is a specific smaller graph.

The study enhances classification of symmetric graphs based on their cores and automorphism groups.

Abstract

A retract of a graph $\Gamma$ is an induced subgraph $\Psi$ of $\Gamma$ such that there exists a homomorphism from $\Gamma$ to $\Psi$ whose restriction to $\Psi$ is the identity map. A graph is a core if it has no nontrivial retracts. In general, the minimal retracts of a graph are cores and are unique up to isomorphism; they are called the core of the graph. A graph $\Gamma$ is $G$-symmetric if $G$ is a subgroup of the automorphism group of $\Gamma$ that is transitive on the vertex set and also transitive on the set of ordered pairs of adjacent vertices. If in addition the vertex set of $\Gamma$ admits a nontrivial partition that is preserved by $G$, then $\Gamma$ is an imprimitive $G$-symmetric graph. In this paper cores of imprimitive symmetric graphs $\Gamma$ of order a product of two distinct primes are studied. In many cases the core of $\Gamma$ is determined completely. In other cases it is proved that either $\Gamma$ is a core or its core is isomorphic to one of two graphs, and conditions on when each of these possibilities occurs is given.