Distinguishing graphs with two integer matrices

TL;DR

This paper investigates how combining the spectrum and Smith normal form of two matrices associated with a graph enhances the ability to distinguish non-isomorphic graphs, providing computational evidence of improved discrimination over previous methods.

Contribution

It introduces a combined approach using spectrum and Smith normal form of two matrices to better distinguish graphs, with extensive computational analysis for graphs up to 10 vertices.

Findings

Number of non-isomorphic graphs with same parameter is less than 100 for 10 vertices.

Combining two matrices' parameters improves graph distinction.

Computational results surpass previous similar studies.

Abstract

It is well known that the spectrum and the Smith normal form of a matrix can be computed in polynomial time. Thus, it is interesting to explore how good are these parameters for distinguishing graphs. This is relevant since it is related to the Graph Isomorphism Problem (GIP), which asks to determine whether two graphs are isomorphic. In this paper, we explore the computational advantages of using the spectrum and the Smith normal form of two matrices associated with a graph. By considering the SNF or the spectrum of two matrices of a graph as a single parameter, we compute the number of non-isomorphic graphs with the same parameter with up to 9 vertices, and with up to 10 vertices when the number of graphs with 9 vertices with the same parameter is less than 1000. Focusing on the best 20 combinations of matrices for graphs with 10 vertices, we notice that the number of such graphs with a mate is less than 100 in any of these 20 cases. This computational result improves on similar previous explorations. Therefore, the use of the spectrum or the SNF of two matrices at the same time shows a substantial improvement in distinguishing graphs.