Note on Generalized Distance Matrix of Graphs

Abdul Hameed, Zia Ullah Khan

Abstract

For any connected graph $G$ with $\alpha \in [0,1]$, the generalized distance matrix $D_{\alpha}(G)$ is defined as, $D_{\alpha}(G)= \alpha T(G)+(1 - \alpha)D(G)$, where $T(G)$ represents a transmission diagonal matrix of~$G$ and $D(G)$ is the distance matrix of~$G$. The maximum eigenvalue of generalized distance matrix is called $D_{\alpha}(G)$ spectral radius. In this paper, we give some new results on generalized distance spectra of complete multipartite graphs. Later in the paper are given some sharp bounds for generalized distance spectral radius of connected graphs.