University of Isfahan Transactions on Combinatorics 2251-8657 15 2 2025 10 13 The relation between distance Laplacian spectral radius and integer $k$-matching number in graphs 125 136 29558 10.22108/toc.2025.143292.2221 EN Yanhong Zhang Department of Mathematics and Statistics, Qinghai Normal University, Xining, China Lei Zhang Department of Mathematics and Statistics, Qinghai Normal University, Xining, China 0000-0001-5187-7898 Haizhen Ren Department of Mathematics and Statistics, Qinghai Normal University, Xining, China 0000-0001-5609-5924 Journal Article 2024 11 05 Let $G$ be a graph with order $n$. Aouchiche and Hansen first proposed the distance Laplacian matrix of $G$, defined as $\mathcal{L}(G)=diag(Tr)-\mathcal{D}(G)$, where $\mathcal{D}(G)$ is the distance matrix and $diag(Tr)=diag(Tr(v_1), Tr(v_2),\ldots,Tr(v_n))$ is the diagonal matrix of the vertex transmissions of $G$, and the largest eigenvalue of $\mathcal{L}(G)$ is called the distance Laplacian spectral radius of $G$, written as $\rho_{\mathcal{L}}(G)$. By using the equitable quotient matrix of $\mathcal{L}(G)$, Tutte Theorem and Tutte-Berge Formula of integer $k$-matching, we establish the lower bound for the distance Laplacian spectral radius of $G$ among all $n$-vertex graphs with given integer $k$-matching number and characterized the corresponding extremal graph. This generalizes the results of Wang et al. [Lower bounds of distance Laplacian spectral radii of $n$-vertex graphs in terms of matching number, <em>Linear Algebra Appl.</em>, <strong>506</strong> (2016) 579--587.] and Liu et al. [Lower bounds of distance Laplacian spectral radii of $n$-vertex graphs in terms of fractional matching number, <em>J. Oper. Res. Soc. China.</em>, (2023) 1--8.]. https://toc.ui.ac.ir/article_29558_3baa38bda30792f662013d4310a8e902.pdf