标题： 连通图的距离和距离无符号拉普拉斯谱差 显示英文标题

标题： Distance and distance signless Laplacian spread of connected graphs

摘要： 对于一个具有$n$个顶点的连通图$G$，回忆一下，$G$的距离无符号拉普拉斯矩阵被定义为$\mathcal{Q}(G)=Tr(G)+\mathcal{D}(G)$，其中$\mathcal{D}(G)$是距离矩阵，$Tr(G)=diag(D_1, D_2, \ldots, D_n)$和$D_{i}$是对应于顶点$v_{i}$的$\mathcal{D}(G)$的行和。 记$\rho^{\mathcal{D}}(G),$ $\rho_{min}^{\mathcal{D}}(G)$ 为$\mathcal{D}(G)$的最大特征值和最小特征值。 并记$q^{\mathcal{D}}(G)$、$q_{min}^{\mathcal{D}}(G)$为$\mathcal{Q}(G)$的最大特征值和最小特征值。 图$G$的距离谱半径定义为$S_{\mathcal{D}}(G)=\rho^{\mathcal{D}}(G)- \rho_{min}^{\mathcal{D}}(G)$，而图$G$的距离无符号拉普拉斯谱半径定义为$S_{\mathcal{Q}}(G)=q^{\mathcal{D}}(G)-q_{min}^{\mathcal{D}}(G)$。 在本文中，我们指出了文献"图的距离谱半径"[G.L. Yu, 等, 离散应用数学。160 (2012) 2474--2478]中定理2.4的结果中的一个错误，并加以修正。 同时，我们得到了图的距离无符号拉普拉斯谱半径的一些下界。 显示英文摘要

摘要： For a connected graph $G$ on $n$ vertices, recall that the distance signless Laplacian matrix of $G$ is defined to be $\mathcal{Q}(G)=Tr(G)+\mathcal{D}(G)$, where $\mathcal{D}(G)$ is the distance matrix, $Tr(G)=diag(D_1, D_2, \ldots, D_n)$ and $D_{i}$ is the row sum of $\mathcal{D}(G)$ corresponding to vertex $v_{i}$. Denote by $\rho^{\mathcal{D}}(G),$ $\rho_{min}^{\mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $\mathcal{D}(G)$, respectively. And denote by $q^{\mathcal{D}}(G)$, $q_{min}^{\mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $\mathcal{Q}(G)$, respectively. The distance spread of a graph $G$ is defined as $S_{\mathcal{D}}(G)=\rho^{\mathcal{D}}(G)- \rho_{min}^{\mathcal{D}}(G)$, and the distance signless Laplacian spread of a graph $G$ is defined as $S_{\mathcal{Q}}(G)=q^{\mathcal{D}}(G)-q_{min}^{\mathcal{D}}(G)$. In this paper, we point out an error in the result of Theorem 2.4 in "Distance spectral spread of a graph" [G.L. Yu, et al, Discrete Applied Mathematics. 160 (2012) 2474--2478] and rectify it. As well, we obtain some lower bounds on ddistance signless Laplacian spread of a graph.