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For a connected graph $G$ on $n$ vertices, recall that the distance signless Laplacian matrix of $G$ is defined to be $mathcal{Q}(G)=Tr(G)+mathcal{D}(G)$, where $mathcal{D}(G)$ is the distance matrix, $Tr(G)=diag(D_1, D_2, ldots, D_n)$ and $D_{i}$ is the row sum of $mathcal{D}(G)$ corresponding to vertex $v_{i}$. Denote by $rho^{mathcal{D}}(G),$ $rho_{min}^{mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $mathcal{D}(G)$, respectively. And denote by $q^{mathcal{D}}(G)$, $q_{min}^{mathcal{D}}(G)$ the largest eigenvalue and the least eigenvalue of $mathcal{Q}(G)$, respectively. The distance spread of a graph $G$ is defined as $S_{mathcal{D}}(G)=rho^{mathcal{D}}(G)- rho_{min}^{mathcal{D}}(G)$, and the distance signless Laplacian spread of a graph $G$ is defined as $S_{mathcal{Q}}(G)=q^{mathcal{D}}(G)-q_{min}^{mathcal{D}}(G)$. In this paper, we point out an error in the result of Theorem 2.4 in Distance spectral spread of a graph [G.L. Yu, et al, Discrete Applied Mathematics. 160 (2012) 2474--2478] and rectify it. As well, we obtain some lower bounds on ddistance signless Laplacian spread of a graph.
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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 $mathcal{D}_{alpha}(G)=alpha Tr(G)+(1-alpha)mathcal{D}(G),~0leqalphaleq1$, 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 $mathcal{D}-$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.
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