Distinguishing number and distinguishing index of neighbourhood corona of two graphs

TL;DR

This paper investigates the automorphisms, distinguishing number, and distinguishing index of the neighbourhood corona of two graphs, providing bounds and structural insights into these graph invariants.

Contribution

It characterizes automorphisms of the neighbourhood corona and derives upper bounds for its distinguishing number and index, advancing understanding of symmetry breaking in complex graph constructions.

Findings

Automorphisms of the neighbourhood corona are characterized.

Upper bounds for the distinguishing number and index are established.

Structural properties of the neighbourhood corona influence its symmetry measures.

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 neighbourhood corona of two graphs $G_1$ and $G_2$ is denoted by $G_1 \star G_2$ and is the graph obtained by taking one copy of $G_1$ and $|V(G_1)|$ copies of $G_2$, and joining the neighbours of the $i$th vertex of $G_1$ to every vertex in the $i$th copy of $G_2$. In this paper we describe the automorphisms of the graph $G_1\star G_2$. Using results on automorphisms, we study the distinguishing number and the distinguishing index of $G_1\star G_2$. We obtain upper bounds for $D(G_1\star G_2)$ and $D'(G_1\star G_2)$.