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

TL;DR

This paper investigates how the distance signless Laplacian spectral radius can be used to determine the existence of perfect matchings in graphs and bipartite graphs, providing new spectral conditions.

Contribution

It introduces two sufficient spectral conditions based on the distance signless Laplacian spectral radius for guaranteeing perfect matchings in graphs and bipartite graphs.

Findings

Derived spectral conditions for perfect matchings

Extended results to bipartite graphs

Provided theoretical bounds for spectral radius

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.