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

Abstract

Let $G$ be a connected $d$-regular graph of order $n$, where $d\ge 3$. Let ${\lambda }_2(G)$ be the second largest eigenvalue of $G$. For even $n$, we show that $G$ contains $\lfloor \frac{2}{3}(d-{\lambda }_2(G))\rfloor$ edge-disjoint perfect matchings. This improves a result stated by Cioabă, 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.