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

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_۱), Tr(v_۲),\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., ۵۰۶ (۲۰۱۶) ۵۷۹--۵۸۷.] 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., (۲۰۲۳) ۱--۸.].

کلیدواژه ها:

graph ، Integer k-matching ، Distance Laplacian ، spectral radius