On automorphisms and fixing number of co-normal product of graphs

TL;DR

This paper investigates the automorphisms and fixing number of co-normal product graphs, providing bounds and exact values for specific graph families, thereby enhancing understanding of graph symmetries and their breaking.

Contribution

It offers new bounds and exact calculations for automorphisms and fixing numbers of co-normal product graphs, a less-explored graph operation.

Findings

Sharp bounds on automorphism group order

Bounds on fixing number of co-normal product graphs

Fixing number computed for specific graph families

Abstract

An automorphism of a graph describes its structural symmetry and the concept of fixing number of a graph is used for breaking its symmetries (except the trivial one). In this paper, we evaluate automorphisms of the co-normal product graph $G_1\ast G_2$ of two simple graphs $G_1$ and $G_2$ and give sharp bounds on the order of its automorphism group. We study the fixing number of $G_1\ast G_2$ and prove sharp bounds on it. Moreover, we compute the fixing number of the co-normal product of some families of graphs.