arXiv 2007.044792020-07math.CO

无符号Laplacian谱半径与图的匹配

Signless Laplacian spectral radius and matching in graphs

Chang Liu, Yingui Pan, Jianping Li

摘要：

图$G$的无符号拉普拉斯矩阵由$Q(G)=D(G)+A(G)$给出，其中$D(G)$是顶点度的对角矩阵，$A(G)$是邻接矩阵。$Q(G)$的最大特征值称为无符号拉普拉斯谱半径，用$q_1=q_1(G)$表示。本文建立了图的无符号拉普拉斯谱半径与完美匹配之间的一些性质。设$r(n)$是方程$x^3-(3n-7)x^2+n(2n-7)x-2(n^2-7n+12)=0$的最大根。我们表明，$G$有一个完美的匹配$n=4$或$n\geq10$，如果$q_1(G)>r(n)$，和$n=6$或$n=8$，如果$q_1(G)>4+2\sqrt{3}$或$q_1(G)>6+2\sqrt{6}$，其中$n$是一个正的偶数。此外，存在图$K_{n-3}\vee K_1 \vee \overline{K_2}$使得$q_1(K_{n-3}\vee K_1 \vee \overline{K_2})=r(n)$是$n\geq4$，存在图$K_2\vee\overline{K_4}$使得$q_1(K_2\vee\overline{K_4})=4+2\sqrt{3}$，存在图$K_3\vee\overline{K_5}$使得$q_1(K_3\vee\overline{K_5})=6+2\sqrt{6}$。这些图都没有完美的匹配。

英文摘要：

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 $n\geq10$, 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 $n\geq4$, 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.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

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