The smallest pair of cospectral cubic graphs with different chromatic indexes

TL;DR

This paper identifies the smallest pair of cospectral cubic graphs with different chromatic indexes, demonstrating that spectral properties do not determine the chromatic index, and explores the limitations of certain graph transformation methods.

Contribution

It provides the first known example of a cospectral pair with different chromatic indexes among cubic graphs of order 16 and analyzes the algebraic properties of their adjacency matrices.

Findings

Found a unique cospectral pair of cubic graphs with different chromatic indexes.

Proved that the orthogonal matrix relating their adjacency matrices cannot be rational.

Showed that this pair cannot be generated by the GM-switching method or similar rational-based techniques.

Abstract

Using an exhaustive search on cubic graphs of order 16, we find a unique cospectral pair with different chromatic indexes. This example indicates that the chromatic index of a regular graph is not characterized by its spectrum, which answers a question recently posed in [O. Etesami, W. H. Haemers, On NP-hard graph properties characterized by the spectrum, Discrete Appl. Math., 285(2020)526-529]. We prove that any orthogonal matrix representing the similarity between the two adjacency matrices of the cospectral pair cannot be rational. This implies that the cospectral pair cannot be obtained using the original GM-switching method or its generalizations based on rational orthogonal matrices.