Signless Laplacian spectral radius and matching in graphs

Abstract

The signless Laplacian matrix of a graph $G$ is given by $Q(G)=D(G)+A(G)$, where $D(G)$ is a diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix. The largest eigenvalue of $Q(G)$ is called the signless Laplacian spectral radius, denoted by $q_1=q_1(G)$. In this paper, some properties between the signless Laplacian spectral radius and perfect matching in graphs are establish. Let $r(n)$ be the largest root of equation $x^3-(3n-7)x^2+n(2n-7)x-2(n^2-7n+12)=0$. We show that $G$ has a perfect matching for $n=4$ or $n\ge 10$, if $q_1(G)>r(n)$, and for $n=6$ or $n=8$, if $q_1(G)>4+2\sqrt{3}$ or $q_1(G)>6+2\sqrt{6}$ respectively, where $n$ is a positive even integer number. Moreover, there exists graphs $K_{n-3}\vee K_1\vee \overline{K_2}$ such that $q_1(K_{n-3}\vee K_1\vee \overline{K_2})=r(n)$ if $n\ge 4$, a graph $K_2\vee \overline{K_4}$ such that $q_1(K_2\vee \overline{K_4})=4+2\sqrt{3}$ and a graph $K_3\vee \overline{K_5}$ such that $q_1(K_3\vee \overline{K_5})=6+2\sqrt{6}$. These graphs all have no prefect matching.

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