On regular graphs with four distinct eigenvalues

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• 作者：Xueyi Huang ^{huangxymath@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Qiongxiang Huang ; ^{huangqx@xju.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：512
• 期：Complete
• 页码：219-233
• 全文大小：375 K

文摘

Let G(4,2)$\mathcal{G}(4,2)$ be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, G(4,2,−1)$\mathcal{G}(4,2,-1)$ (resp. G(4,2,0)$\mathcal{G}(4,2,0)$) the set of graphs belonging to G(4,2)$\mathcal{G}(4,2)$ with −1 (resp. 0) as an eigenvalue, and G(4,≥−1)$\mathcal{G}(4,\ge -1)$ the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than −1. In this paper, we prove the non-existence of connected graphs having four distinct eigenvalues in which at least three eigenvalues are simple, and determine all the graphs in G(4,2,−1)$\mathcal{G}(4,2,-1)$. As a by-product of this work, we characterize all the graphs belonging to G(4,≥−1)$\mathcal{G}(4,\ge -1)$ and G(4,2,0)$\mathcal{G}(4,2,0)$, respectively, and show that all these graphs are determined by their spectra.