Cospectral regular graphs with and without a perfect matching

TL;DR

This paper constructs pairs of cospectral regular graphs with the same degree where one has a perfect matching and the other does not, addressing a previously open problem in graph theory.

Contribution

It provides explicit constructions of cospectral regular graphs differing in the existence of perfect matchings, solving a known research problem.

Findings

Existence of cospectral regular graphs with and without perfect matchings for all degrees ≥ 5

Explicit construction methods for such graph pairs

Resolution of a problem posed at the British Combinatorial Conference

Abstract

For each $b \geq 5$ we construct a pair of cospectral $b$-regular graphs, where one has a perfect matching and the other one not. This solves a research problem posed by the third author at the 22nd British Combinatorial Conference.