论文标题
无标志性的拉普拉斯光谱半径和图形匹配
Signless Laplacian spectral radius and matching in graphs
论文作者
论文摘要
图$ g $的无标志性拉普拉斯矩阵由$ q（g）= d（g）+a（g）$给出，其中$ d（g）$是顶点度的对角线矩阵，$ a（g）$是邻接矩阵。 $ q（g）$的最大特征值称为无标志性的laplacian光谱半径，用$ q_1 = q_1（g）$表示。在本文中，建立了无标识的拉普拉斯光谱半径与完美匹配之间的某些特性。令$ r（n）$为方程的最大根$ x^3-（3n-7）x^2+n（2n-7）x-2（n^2-7n+12）= 0 $。我们表明，$ g $具有$ n = 4 $或$ n \ geq10 $的完美匹配，如果$ q_1（g）> r（n）$，以及$ n = 6 $或$ n = 8 $，如果$ q_1（g）> 4+2 \ 2 \ sqrt {3} $或$ q_1（g）$ q_1（g）> 6+2 $ n $ n n $ n n $ n n $ n ander in $ n} 数字。此外，存在图形$ k_ {n-3} \ vee k_1 \ vee \ overline {k_2} $，使得$ q_1（k_ {n-3} \ vee k_1 \ vee k_1 \ vee \ vee \ edimline {k_2}}这样的$ q_1（k_2 \ vee \ overline {k_4}）= 4+2 \ 2 \ sqrt {3} $和图$ k_3 \ vee \ overline {k_5} $，使得$ q_1（$ q_1）这些图都没有县匹配。
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.
