Distinguishing number and distinguishing index of certain graphs

TL;DR

This paper calculates the distinguishing number and index for specific graphs and explores these parameters in the corona product of two graphs, advancing understanding of graph symmetry breaking.

Contribution

It provides explicit computations of the distinguishing number and index for certain graphs and analyzes their behavior in the corona product operation.

Findings

Computed $D(G)$ and $D'(G)$ for specific graphs.

Analyzed how $D(G)$ and $D'(G)$ behave under the corona product.

Extended the understanding of symmetry breaking in graph products.

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. In this paper we compute these two parameters for some specific graphs. Also we study the distinguishing number and the distinguishing index of corona product of two graphs.