The Asymmetric Index of a Graph

TL;DR

This paper introduces the asymmetric index of a graph, a measure of how many edge modifications are needed to transform a non-asymmetric graph into an asymmetric one, extending previous concepts of graph asymmetry.

Contribution

The paper defines the asymmetric index as a new metric quantifying the minimal edge changes required to achieve graph asymmetry, expanding on earlier work by Erdős and Rényi.

Findings

Introduced the asymmetric index as a new graph property

Provided methods to compute the asymmetric index for various graphs

Established bounds and properties of the asymmetric index

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.