Eigenvalues, edge-disjoint perfect matchings and toughness of regular graphs

TL;DR

This paper establishes new bounds on the number of edge-disjoint perfect matchings in regular graphs based on eigenvalues and provides conditions on the second eigenvalue to ensure the graph's toughness exceeds one.

Contribution

It improves existing bounds on perfect matchings and toughness in regular graphs by relating eigenvalues to structural properties.

Findings

Number of edge-disjoint perfect matchings depends on eigenvalue gap

Sharp upper bounds on second eigenvalue guarantee toughness > 1

Enhanced understanding of spectral conditions for graph robustness

Abstract

Let $G$ be a connected $d$-regular graph of order $n$, where $d\geq3$. Let $\lambda_{2}(G)$ be the second largest eigenvalue of $G$. For even $n$, we show that $G$ contains $\left\lfloor\frac{2}{3}(d-\lambda_{2}(G))\right\rfloor$ edge-disjoint perfect matchings. This improves a result stated by Cioab\u{a}, Gregory and Haemers \cite{CGH}. Let $t(G)$ be the toughness of $G$. When $G$ is non-bipartite, we give a sharp upper bound of $\lambda_{2}(G)$ to guarantee that $t(G)>1$. This enriches the previous results on this direction.