Discrimination of Graph Isomorphism Classes by Continuous Spectrum and Split Technique

TL;DR

This paper introduces a simple, efficient spectral method based on a new matrix representation to distinguish non-isomorphic graphs, especially effective for strongly regular graphs, with practical time complexity.

Contribution

A novel eigenvalue-based technique that reliably discriminates non-isomorphic graphs without heuristics, improving practical efficiency for challenging graph classes.

Findings

Almost always separates non-isomorphic graphs in O(n^3) time

Successfully characterizes isomorphism classes of strongly regular graphs up to 64 vertices

Effective even for strongly regular graphs, considered hard cases

Abstract

The graph isomorphism problem is a main problem which has numerous applications in different fields. Thus, finding an efficient and easy to implement method to discriminate non-isomorphic graphs is valuable. In this paper, a new method is introduced which is very simple and easy to implement, but very efficient in discriminating non-isomorphic graphs, in practice. This method does not need any heuristic attempt and based on the eigenvalues of a new matrix representation for graphs. It, almost always, separates non-isomorphic $n$-vertex graphs in time $O(n^3)$ and in worst cases such as strongly regular graphs, in time $O(n^4)$. Here, we show that this method, successfully, characterizes the isomorphism classes of studied instances of strongly regular graphs (up to 64 vertices). Strongly regular graphs are believed to be hard cases of the graph isomorphism problem.