ABSTRACT

Let ${\mathrm{\Re }}_{n,\text{β}}$ be the family of graphs with n vertices and with the matching number β. The distance signless Laplacian spectral radius of a graph G is denoted by $\rho \left(G\right)$. Li and Sun (Matching number, connectivity and eigenvalues of distance signless Laplacians. Linear Multilinear Algebra, 2019. https://doi.org/10.1080/03081087.2019.1588218) proved that if $n\ge \frac{7}{2}\text{β}$, $\rho \left(G\right)\ge \rho (K_{\text{β}}\bigvee \overline{K_{n-\text{β}}})$ for any $G\in {\mathrm{\Re }}_{n,\text{β}}$, with equality if and only if $G=K_{\text{β}}\bigvee \overline{K_{n-\text{β}}}$. Further Li and Sun conjectured that if $2\text{β}+2\le n\le \frac{7}{2}\text{β}$, the statement holds. In this paper, we disprove this conjecture by an example, when n = 19 and $\text{β}=8$. Further by using a different technique, we show that the statement holds for $n\ge \frac{25}{8}\text{β}$.

摘要

让${\mathrm{\Re }}_{n,\text{β}}$是具有n个顶点和匹配数β的图族。图G的距离无符号拉普拉斯谱半径表示为$\rho \left(G\right)$. Li 和 Sun（Matching number, connectivity and eigenvalues of distanceless Laplacians. Linear Multilinear Algebra, 2019. https://doi.org/10.1080/03081087.2019.1588218）证明如果$n\ge \frac{7}{2}\text{β}$,$\rho \left(G\right)\ge \rho ({ķ}_{\text{β}}\bigvee \overline{{ķ}_{n-\text{β}}})$对于任何$G\in {\mathrm{\Re }}_{n,\text{β}}$, 相等当且仅当$G={ķ}_{\text{β}}\bigvee \overline{{ķ}_{n-\text{β}}}$. Li 和 Sun 进一步推测，如果$2\text{β}+2\le n\le \frac{7}{2}\text{β}$，该陈述成立。在本文中，我们通过一个例子来反驳这个猜想，当n  = 19 并且$\text{β}=8$. 进一步通过使用不同的技术，我们证明该陈述适用于$n\ge \frac{25}{8}\text{β}$.