Note on distance signless Laplacian spectral radius under given matching number

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 $n\ge \frac{25}{8}\text{β}$.

Keywords:

• Distance signless Laplacian spectral radius
• matching number

2010 Mathematics Subject Classification:

• 05C50

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by the Science and Technology Program of Guangzhou, China “Research on the generalized matching theory and application of statistical model” (authorized in 2019) and by the Qinghai Province Natural Science Foundation (No. 2020-ZJ-924).