Distance matrix correlation spectrum of graphs

Abstract

Let $G$ be a simple, connected graph, $\mathcal{D}(G)$ be the distance matrix of $G$, and $Tr(G)$ be the diagonal matrix of vertex transmissions of $G$. The distance Laplacian matrix and distance signless Laplacian matrix of $G$ are defined by $\mathcal{L}(G) = Tr(G)-\mathcal{D}(G)$ and $\mathcal{Q}(G) = Tr(G)+\mathcal{D}(G)$, respectively. The eigenvalues of $\mathcal{D}(G)$, $\mathcal{L}(G)$ and $\mathcal{Q}(G)$ is called the $\mathcal{D}-$spectrum, $\mathcal{L}-$spectrum and $\mathcal{Q}-$spectrum, respectively. The generalized distance matrix of $G$ is defined as $\mathcal{D}_α(G)=αTr(G)+(1-α)\mathcal{D}(G),~0\leqα\leq1$, and the generalized distance spectral radius of $G$ is the largest eigenvalue of $\mathcal{D}_α(G)$. In this paper, we give a complete description of the $\mathcal{D}-$spectrum, $\mathcal{L}-$spectrum and $\mathcal{Q}-$spectrum of some graphs obtained by operations. In addition, we present some new upper and lower bounds on the generalized distance spectral radius of $G$ and of its line graph $L(G)$, based on other graph-theoretic parameters, and characterize the extremal graphs. Finally, we study the generalized distance spectrum of some composite graphs.