Distance signless Laplacian spectral radius and perfect matching in graphs and bipartite graphs

Abstract

The distance matrix $\mathcal{D}$ of a connected graph $G$ is the matrix indexed by the vertices of $G$ which entry $\mathcal{D}_{i,j}$ equals the distance between the vertices $v_i$ and $v_j$. The distance signless Laplacian matrix $\mathcal{Q}(G)$ of graph $G$ is defined as $\mathcal{Q}(G)=Diag(Tr)+\mathcal{D}(G)$, where $Diag(Tr)$ is the diagonal matrix of the vertex transmissions in $G$. The largest eigenvalue of $\mathcal{Q}(G)$ is called the distance signless Laplacian spectral radius of $G$, written as $\eta_1(G)$. And a perfect matching in a graph is a set of disadjacent edges covering every vertex of $G$. In this paper, we present two suffcient conditions in terms of the distance signless Laplacian sepectral radius for the exsitence of perfect matchings in graphs and bipatite graphs.