The relation between distance Laplacian spectral radius and integer $k$-matching number in graphs

Document Type : Research Paper

Authors

Department of Mathematics and Statistics, Qinghai Normal University, Xining, China

Abstract

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, Linear Algebra Appl., 506 (2016) 579--587.] and Liu et al. [Lower bounds of distance Laplacian spectral radii of $n$-vertex graphs in terms of fractional matching number, J. Oper. Res. Soc. China., (2023) 1--8.].

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References

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History

  • Receive Date: 05 November 2024
  • Revise Date: 19 May 2025
  • Accept Date: 29 May 2025
  • Published Online: 13 October 2025

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