Spectral characterization of matchings in graphs

TL;DR

This paper provides a spectral characterization of the maximum matching size in graphs, linking graph theory with spectral properties of skew-symmetric matrices.

Contribution

It introduces a spectral criterion for identifying graphs with a given matching number based on skew rank and spectrum realization.

Findings

Graphs with maximum skew rank 2k have matching number k.

For any set of k nonzero purely imaginary numbers, a real skew-symmetric matrix with the graph and spectrum exists.

The matching number is characterized by spectral properties of associated matrices.

Abstract

A spectral characterization of the matching number (the size of a maximum matching) of a graph is given. More precisely, it is shown that the graphs G of order n whose matching number is k are precisely those graphs with the maximum skew rank 2k such that for any given set of k distinct nonzero purely imaginary numbers there is a real skew-symmetric matrix A with graph G whose spectrum consists of the given k numbers, their conjugate pairs, and n-2k zeros.