ABSTRACT

For a simple connected graph G, let $D\left(G\right)$, $Tr\left(G\right)$, $D^L\left(G\right)$ and $D^Q\left(G\right)$, respectively, are the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix. The generalized distance matrix $D_{\alpha }\left(G\right)$ of G is the convex linear combinations of $Tr\left(G\right)$ and $D\left(G\right)$ and is defined as $D_{\alpha }\left(G\right)=\alpha Tr\left(G\right)+(1-\alpha )D\left(G\right)$, for $0\le \alpha \le 1$. As $D_0\left(G\right)=D\left(G\right),2D_{\frac{1}{2}}\left(G\right)=D^Q\left(G\right),D_1\left(G\right)=Tr\left(G\right)$ and $D_{\alpha }\left(G\right)-D_{\text{β}}\left(G\right)=(\alpha -\text{β})D^L\left(G\right)$, this matrix reduces to merging the distance spectral and distance signless Laplacian spectral theories. Let ${\mathrm{\partial }}_1\left(G\right)\ge {\mathrm{\partial }}_2\left(G\right)\ge \cdots \ge {\mathrm{\partial }}_n\left(G\right)$ be the eigenvalues of $D_{\alpha }\left(G\right)$ and let $S_{D_{\alpha }}\left(G\right)={\mathrm{\partial }}_1\left(G\right)-{\mathrm{\partial }}_n\left(G\right)$ be the generalized distance spectral spread of the graph G. In this paper, we obtain bounds for the generalized distance spectral spread $S_{D_{\alpha }}\left(G\right)$. We also obtain a relation between the generalized distance spectral spread $S_{D_{\alpha }}\left(G\right)$ and the distance spectral spread $S_D\left(G\right)$. Further, we obtain lower bounds for $S_{D_{\alpha }}\left(G\right)$ of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for $S_{D_{\alpha }}\left(G\right)$ in terms of clique number and independence number of the graph G and characterize the extremal graphs for some cases.

摘要

对于一个简单的连通图G，让$D\left(G\right)$,$吨r\left(G\right)$,$D^{\mathrm{大号}}\left(G\right)$和$D^{问}\left(G\right)$，分别是距离矩阵、顶点传输的对角矩阵、距离拉普拉斯矩阵和距离无符号拉普拉斯矩阵。广义距离矩阵$D_{\alpha }\left(G\right)$G的凸线性组合$吨r\left(G\right)$和$D\left(G\right)$并定义为$D_{\alpha }\left(G\right)=\alpha 吨r\left(G\right)+(1-\alpha )D\left(G\right)$， 为了$0\le \alpha \le 1$. 作为$D_0\left(G\right)=D\left(G\right),2D_{\frac{1}{2}}\left(G\right)=D^{问}\left(G\right),D_1\left(G\right)=吨r\left(G\right)$和$D_{\alpha }\left(G\right)-D_{\text{β}}\left(G\right)=(\alpha -\text{β})D^{\mathrm{大号}}\left(G\right)$，该矩阵简化为合并距离谱和距离无符号拉普拉斯谱理论。让${\mathrm{\partial }}_1\left(G\right)\ge {\mathrm{\partial }}_2\left(G\right)\ge \cdots \ge {\mathrm{\partial }}_n\left(G\right)$是的特征值$D_{\alpha }\left(G\right)$然后让${\mathrm{小号}}_{D_{\alpha }}\left(G\right)={\mathrm{\partial }}_1\left(G\right)-{\mathrm{\partial }}_n\left(G\right)$是图G的广义距离谱扩展。在本文中，我们获得了广义距离谱扩展的界限${\mathrm{小号}}_{D_{\alpha }}\left(G\right)$. 我们还获得了广义距离谱扩展之间的关系${\mathrm{小号}}_{D_{\alpha }}\left(G\right)$和距离谱扩展${\mathrm{小号}}_D\left(G\right)$. 此外，我们获得了下界${\mathrm{小号}}_{D_{\alpha }}\left(G\right)$涉及不同图参数的二部图，我们描述了某些情况下的极值图。我们还获得了下界${\mathrm{小号}}_{D_{\alpha }}\left(G\right)$在图G的团数和独立数方面，并在某些情况下刻画极值图。