The matching number of G, written α′(G)${\alpha }^{\prime }(G)$${\alpha }^{\prime }(G)$, is the size of a maximum matching in G. Suppose that n and k are positive integers of the same parity. Let θ′(n,k)${\theta }^{\prime }(n,k)$ be the largest root of $x^3-(n-k-2)x^2-(n-1)x+k(n-k-2)=0$ and $\theta (n,k)=\{\begin{array}{ll}{\theta }^{\prime }(n,k)& \mathrm{if}n\ge 3k+2\\ {\displaystyle \frac{n-k-2+\sqrt{(n-k-2{)}^2+4(n^2-k^2)}}{4}}& \mathrm{if}n\le 3k\end{array}.$In this article, we prove that for a positive integer $n\ge k+2$, if G is an n-vertex connected graph with the spectral radius $\rho (G)>\theta (n,k)$, then ${\alpha }^{\prime }(G)>\frac{n-k}{2}$. The bound is sharp in the sense that for every positive integer $n\ge k+2$, there are graphs H with $\rho (H)=\theta (n,k)$ and ${\alpha }^{\prime }(G)=\frac{n-k}{2}$.

中文翻译：

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光谱半径图中的匹配

G的匹配数，写为α′( G )${\alpha }^{\prime }（G）$${\alpha }^{\prime }(G)$，是G中最大匹配的大小。假设n和k是相同奇偶校验的正整数。让θ ′ ( n , k )${\theta }^{\prime }(n,k)$为最大根$X^3-（n-k-2）X^2-（n-1）X+k（n-k-2）=0$和$\theta （n,k）=\{\begin{array}{ll}{\theta }^{\prime }（n,k）& \mathrm{如果}n\ge 3k+2\\ {\displaystyle \frac{n-k-2+\sqrt{（n-k-2{）}^2+4（n^2-k^2）}}{4}}& \mathrm{如果}n\le 3k\end{array}。$在本文中，我们证明对于正整数$n\ge k+2$，如果G是具有谱半径的n顶点连通图$\rho （G）>\theta （n,k）$， 然后${\alpha }^{\prime }（G）>\frac{n-k}{2}$。界限是尖锐的，因为对于每个正整数$n\ge k+2$，有图H与$\rho （H）=\theta （n,k）$和${\alpha }^{\prime }（G）=\frac{n-k}{2}$。