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arXiv · 1808.10467

The Asymmetric Index of a Graph

Alejandra Brewer·Adam Gregory·Quindel Jones·Darren A. Narayan
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Abstract

A graph $G$ is asymmetric if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erd\H{o}s and R\'{e}nyi in 1963 where they measured the degree of asymmetry of an asymmetric graph. They proved that any asymmetric graph can be made non-asymmetric by removing some number $r$ of edges and/or adding adding some number $s$ of edges, and defined the degree of asymmetry of a graph to be the minimum value of $r+s$. In this paper, we define another property that how close a given non-asymmetric graph is to being asymmetric. We define the asymmetric index of a graph $G$, denoted $ai(G)$, to be the minimum of $r+s$ in order to change $G$ into an asymmetric graph.

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Alejandra Brewer, Adam Gregory, Quindel Jones, Darren A. Narayan. 2018-08-30. The Asymmetric Index of a Graph. https://arxiv.org/abs/1808.10467

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