Let $r$ and $m$ be two integers such that $rgeq m$. Let $H$ be a graph with order $|H|$, size $e$ and maximum degree $r$ such that $2egeq |H|r-m$. We find a best lower bound on spectral radius of graph $H$ in terms of $m$ and $r$. Let $G$ be a connected $r$-regular graph of order $|G|$ and $ k r$ be an integer. Using the previous results, we find some best upper bounds (in terms of $r$ and $k$) on the third largest eigenvalue that is sufficient to guarantee that $G$ has a $k$-factor when $k|G|$ is even. Moreover, we find a best bound on the second largest eigenvalue that is sufficient to guarantee that $G$ is $k$-critical when $k|G|$ is odd. Our results extend the work of Cioab?, Gregory and Haemers [J. Combin. Theory Ser. B, 1999] who obtained such results for 1-factors.
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机译:令$ r $和$ m $是两个整数,使得$ r geq m $。假设$ H $是一个图形,其阶次为$ | H | $,大小为$ e $,最大度为$ r $,使得$ 2e geq | H | r-m $。我们找到了图$ H $的光谱半径的最佳下限,分别为$ m $和$ r $。令$ G $为阶数$ | G | $的连通$ r $正则图,而$ k 展开▼