Distinguishing number and distinguishing index of Kronecker product of two graphs

TL;DR

This paper investigates the minimum number of labels needed to uniquely identify vertices and edges of the Kronecker product of two graphs, advancing understanding of graph symmetries and automorphisms.

Contribution

It introduces new bounds and properties for the distinguishing number and index specifically for the Kronecker product of graphs, a less-explored graph operation.

Findings

Derived bounds for the distinguishing number of Kronecker products.

Established relationships between the automorphisms of component graphs and their Kronecker product.

Provided examples illustrating the application of the theoretical results.

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex labeling (edge labeling) with $d$ labels that is preserved only by a trivial automorphism. The Kronecker product $G\times H$ of two graphs $G$ and $H$ is the graph with vertex set $V (G)\times V (H)$ and edge set $\{\{(u, x), (v, y)\} | \{u, v\} \in E(G) ~and ~\{x, y\} \in E(H)\}$. In this paper we study the distinguishing number and the distinguishing index of Kronecker product of two graphs.