On spectral spread of generalized distance matrix of a graph

Abstract

For a simple connected graph $G$, let $D(G)$, $Tr(G)$, $D^{L}(G)$ and $D^{Q}(G)$, respectively be the distance matrix, the diagonal matrix of the vertex transmissions, distance Laplacian matrix and the distance signless Laplacian matrix of a graph $G$. The convex linear combinations $D_{\alpha}(G)$ of $Tr(G)$ and $D(G)$ is defined as $D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)$, $0\leq \alpha\leq 1$. As $D_{0}(G)=D(G), ~~~ 2D_{\frac{1}{2}}(G)=D^{Q}(G), ~~~ D_{1}(G)=Tr(G)$ and $D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)$, this matrix reduces to merging the distance spectral, distance Laplacian spectral and distance signless Laplacian spectral theories. Let $\partial_{1}(G)\geq \partial_{2}(G)\geq \dots \geq \partial_{n}(G)$ be the eigenvalues of $D_{\alpha}(G)$ and let $D_{\alpha}S(G)=\partial_{1}(G)-\partial_{n}(G)$ be the generalized distance spectral spread of the graph $G$. In this paper, we obtain some bounds for the generalized distance spectral spread $D_{\alpha}(G)$. We also obtain relation between the generalized distance spectral spread $D_{\alpha}(G)$ and the distance spectral spread $S_{D}(G)$. Further, we obtain the lower bounds for $D_{\alpha}S(G)$ of bipartite graphs involving different graph parameters and we characterize the extremal graphs for some cases. We also obtain lower bounds for $D_{\alpha}S(G)$ in terms of clique number and independence number of the graph $G$ and characterize the extremal graphs for some cases.