The distinguishing index of graphs with at least one cycle is not more than its distinguishing number

TL;DR

This paper investigates the relationship between the distinguishing number and index of graphs with cycles, proving that for such graphs, the index does not exceed the number, and characterizes all graphs where this inequality holds.

Contribution

It establishes that for connected graphs with cycles, the distinguishing index is at most the distinguishing number, and provides a complete characterization of graphs satisfying this condition.

Findings

For graphs with cycles, D'(G) ≤ D(G).

Characterization of all connected graphs with D'(G) ≤ D(G).

Extension of known results from trees to graphs with cycles.

Abstract

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$ we have $D'(G) \leq D(G) + 1$. The complete characterization of finite trees $T$ with $D'(T)=D(T)+ 1$ has been given recently. In this note we show that if $G$ is a finite connected graph with at least one cycle, then $D'(G)\leq D(G)$. Finally, we characterize all connected graphs for which $D'(G) \leq D(G)$.