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

Document Type: Research Paper

Authors

1 Department of Mathematics‎, ‎Yazd University‎, ‎89195-741‎, ‎Yazd‎, ‎Iran.

2 Department of Mathematics‎, ‎Yazd University‎, ‎Yazd‎, ‎Iran.

Abstract

‎The distinguishing number (resp. index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$‎
‎such that $G$ has an vertex labeling (resp. 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 in $G$.‎
‎ The fractional power of $G$‎, ‎is the $m^{th}$ power of the $n$-subdivision of $G$‎, ‎i.e.‎, ‎$(G^{\frac{1}{n}})^m$ or $n$-subdivision of $m$-th power of $G$‎, ‎i.e.‎, ‎$(G^m)^{\frac{1}{n}}$‎. ‎In this paper we study the distinguishing number and the distinguishing index of the natural and the fractional powers of $G$‎. ‎We show that the natural powers more than one of a graph are distinguished by at most three edge labels‎. ‎We also show that for a connected graph $G$ of order $n \geqslant 3$ with maximum degree $\Delta (G)$‎, ‎and for $k\geqslant 2$‎, ‎$D(G^{\frac{1}{k}})\leqslant \lceil \sqrt[k]{\Delta (G)} \rceil$‎. ‎Finally we prove that for $m\geqslant 2$‎, ‎the fractional power of $G$‎, ‎i.e.‎, ‎$(G^{\frac{1}{k}})^m$ and $(G^m)^{\frac{1}{k}}$ are distinguished‎ ‎ by at most three edge labels‎.

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Main Subjects

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History

• Receive Date: 04 October 2016
• Revise Date: 18 September 2017
• Accept Date: 19 September 2017