The Asymmetric Index of a Graph

A graph $G$ is asymmetric if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erdős and Rényi 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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