Distinguishing number and distinguishing index of natural and fractional powers of graphs

TL;DR

This paper investigates the distinguishing number and index of natural and fractional powers of graphs, providing bounds and showing that powers greater than two are distinguishable with three labels.

Contribution

It introduces bounds for the distinguishing number and index of fractional and natural powers of graphs, extending understanding of graph symmetries in these complex structures.

Findings

Natural powers greater than two are distinguished by three labels.

Bounds for the distinguishing number of fractional powers are established.

The distinguishing index of fractional powers is at most three.

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. For any $n \in \mathbb{N}$, the $n$-subdivision of $G$ is a simple graph $G^{\frac{1}{n}}$ which is constructed by replacing each edge of $G$ with a path of length $n$. The $m^{th}$ power of $G$, is a graph with same set of vertices of $G$ and an edge between two vertices if and only if there is a path of length at most $m$ between them. The fractional power of $G$, denoted by $G^{\frac{m}{n}}$ is $m^{th}$ power of the $n$-subdivision of $G$ or $n$-subdivision of $m$-th power of $G$. In this paper we study the distinguishing number and distinguishing index of natural and fractional powers of $G$. We show that the natural powers more than two of a graph distinguished by three edge labels. Also we show that for a connected graph $G$ of order $n \geqslant 3$ with maximum degree $\Delta (G)$, $D(G^{\frac{1}{k}})\leqslant min\{s: 2^k+\sum^s_{n=3}n^{k-1}\geqslant \Delta (G)\}$ and for $m\geqslant 3$, $D'(G^{\frac{m}{k}})\leqslant 3$.