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

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.