Distance matrix correlation spectrum of graphs

arXiv2004.01557MaRDI QIDQ6337994

Wenzhi Liu, Pengli Lu

Publication date: 1 April 2020

Abstract: Let $G$ be a simple, connected graph, $mathcalD(G)$ 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 $mathcalL(G)=Tr(G)-mathcalD(G)$ and $mathcalQ(G)=Tr(G)+mathcalD(G)$, respectively. The eigenvalues of $mathcalD(G)$, $mathcalL(G)$ and $mathcalQ(G)$ is called the $mathcalD-$spectrum, $mathcalL-$spectrum and $mathcalQ-$spectrum, respectively. The generalized distance matrix of $G$ is defined as $mathcal{D}_{alpha}(G)=alphaTr(G)+(1-alpha)mathcalD(G),0leqalphaleq1$, and the generalized distance spectral radius of $G$ is the largest eigenvalue of $mathcal{D}_{alpha}(G)$. In this paper, we give a complete description of the $mathcalD-$spectrum, $mathcalL-$spectrum and $mathcalQ-$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.

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