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arXiv · 2410.04413

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

Wenqian Zhang
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Abstract

Let $G$ be a connected $d$-regular graph of order $n$, where $d\geq3$. Let $λ_{2}(G)$ be the second largest eigenvalue of $G$. For even $n$, we show that $G$ contains $\left\lfloor\frac{2}{3}(d-λ_{2}(G))\right\rfloor$ 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 $λ_{2}(G)$ to guarantee that $t(G)>1$. This enriches the previous results on this direction.

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Wenqian Zhang. 2024-10-06. Eigenvalues, edge-disjoint perfect matchings and toughness of regular graphs. https://arxiv.org/abs/2410.04413

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