ON GENERALIZED DISTANCE SPECTRAL RADIUS OF A BIPARTITE 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, the distance Laplacian matrix and the distance signless Laplacian matrix of a graph $G$. The convex linear combination $D_{\alpha }(G)$ of $Tr(G)$ and $D(G)$ is defined as $D_{\alpha }(G)=\alpha Tr(G)+(1-\alpha )D(G)$, $0\le \alpha \le 1$. As $D_0(G)=D(G)$, $2D_{\frac{1}{2}}(G)=D^Q(G)$, $D_1(G)=Tr(G)$, this matrix reduces to merging the distance spectral, signless distance Laplacian spectral theories. In this paper, we study the spectral radius of the generalized distance matrix $D_{\alpha }(G)$ of a graph $G$. We obtain bounds for the generalized distance spectral radius of a bipartite graph in terms of various parameters associated with the structure of the graph and characterize the extremal graphs. For $\alpha =0$, our results improve some previously known bounds.