Distinguishing number and distinguishing index of join of two graphs

TL;DR

This paper investigates the distinguishing number and index of the join of two graphs, establishing bounds and specific values, thereby advancing understanding of graph symmetries in combined graph structures.

Contribution

It provides bounds for the distinguishing number and index of graph joins and characterizes these parameters for certain classes of joined graphs.

Findings

Bounded the difference between $D(G+H)$ and $ ext{max}igrace D(G), D(H) igrace$

Proved that for connected graphs of order at least 2, the distinguishing index of the join is 2, except for $K_2+K_2$ which is 3

Established specific values and bounds for the distinguishing parameters of joined graphs.

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 study the distinguishing number and the distinguishing index of join of two graphs $G$ and $H$, i.e., $G+H$. We prove that $0\leq D(G+H)-max\{D(G),D(H)\}\leq z$, where $z$ is depends of the number of some induced subgraphs generated by some suitable partitions of $V(G)$ and $V(H)$. Also, we prove that if $G$ is a connected graph of order $n \geq 2$, then $D'(G+ \cdots +G)=2$, except $D'(K_2+K_2)=3$.