On perfect matchings, edge-colourings and eigenvalues of cubic graphs

TL;DR

This paper explores whether the spectrum of a cubic graph's adjacency matrix reveals the presence of perfect matchings, providing counterexamples, extending them, and proposing a new spectral condition for perfect matchings.

Contribution

It introduces infinite families of cospectral cubic graphs with different edge-chromatic numbers and proposes a new spectral criterion for the existence of perfect matchings.

Findings

Counterexamples extend to infinite families via truncation.

Infinite cospectral cubic graphs with different edge-chromatic numbers are constructed.

A new spectral condition for perfect matchings in cubic graphs is established.

Abstract

We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that is, an edge colouring with three colors) the answer is known to be negative. In the latter case, a few counter examples (found by computer) are known. Here we show that these counter examples can be extended to an infinite family by use of truncation. Thus we obtain infinitely many pairs of cospectral cubic graphs with different edge-chromatic number. For all these pairs both graphs have a perfect matching, and the mentioned question is still open. But we do find a new sufficient condition for a perfect matching in a cubic graphs in terms of its spectrum. In addition we obtain a few more results concerning spectral characterizations of cubic graphs.