Regular factors of regular graphs from eigenvalues

TL;DR

This paper establishes bounds on eigenvalues of regular graphs that guarantee the existence of certain factors or criticality, advancing spectral graph theory by linking eigenvalues to graph structure.

Contribution

It provides new bounds on eigenvalues of regular graphs that ensure the presence of k-factors or k-criticality, depending on the parity of the product nk.

Findings

Bound on third largest eigenvalue for even kn ensuring k-factor

Bound on second largest eigenvalue for odd nk ensuring k-criticality

Improved spectral conditions for regular graph properties

Abstract

Let m and r be two integers. Let G be a connected r-regular graph of order n and k an integer depending on m and r. For even kn, we find a best upper bound (in terms of r and m) on the third largest eigenvalue that is sufficient to guarantee that G has a k-factor. When nk is odd, we give a best upper bound (in terms of r and m) on the second largest eigenvalue that is sufficient to guarantee that G is k-critical.