The Asymmetric Index of a Graph
Alejandra Brewer, Adam Gregory, Quindel Jones, and Darren A. Narayan

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 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 of edges and/or adding adding some number of edges, and defined the degree of asymmetry of a graph to be the minimum value of . 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 , denoted , to be the minimum of in order to change into an asymmetric graph.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGraph theory and applications · Advanced Graph Theory Research
