图的无符号拉普拉斯和距离无符号拉普拉斯特征值

A Note on the Signless Laplacian and Distance Signless Laplacian Eigenvalues of Graphs

中文关键词:  无符号拉普拉斯  距离无符号拉普拉斯  谱半径  特征值.

英文关键词:signless Laplacian  distance signless Laplacian  spectral radius  eigenvalues.

中文摘要:

      对于简单图$G$, 我们得到$\delta_i\geq d_i-\sqrt{\lfloor \frac{i}{2} \rfloor \lceil \frac{i}{2} \rceil}$, 其中$\delta_i$和$d_i$分别表示第$i$大无符号拉普拉斯特征值和第$i$ 大顶点的度.通过对简单连通图删除某些点和边, 我们得到关于图的距离无符号拉普拉斯特征值的若干不等式. 此外, 对于图的距离无符号拉普拉斯谱半径, 我们刻画了在给定直径的树以及给定围长的单圈图和双圈图中, 具有最小距离无符号拉普拉斯谱半径的极图.

英文摘要:

      Let $G$ be a simple graph. We first show that $ \delta_i\geq d_i-\sqrt{\lfloor \frac{i}{2} \rfloor \lceil \frac{i}{2} \rceil}$, where $\delta_i$ and $d_i$ denote the $i$-th signless Laplacian eigenvalue and the $i$-th degree of vertex in $G$, respectively. Suppose $G$ is a simple and connected graph, then some inequalities on the distance signless Laplacian eigenvalues are obtained by deleting some vertices and some edges from $G$. In addition, for the distance signless Laplacian spectral radius $\rho_{\mathcal{Q}}(G)$, we determine the extremal graphs with the minimum $\rho_{\mathcal{Q}}(G)$ among the trees with given diameter, the unicyclic and bicyclic graphs with given girth, respectively.