On Generalized Distance Gaussian Estrada Index of Graphs

For a simple undirected connected graph G of order n, let D(G) , DL(G) , DQ(G) and Tr(G) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. The generalized distance matrix Dα(G) is signified by Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1]. Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let ∂1,∂2,…,∂n be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index Pα(G) , as Pα(G)=∑ni=1e−∂2i. Since characterization of Pα(G) is very appealing in quantum information theory, it is interesting to study the quantity Pα(G) and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter α . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index Pα(G) of a connected graph G, involving the different graph parameters, including the order n, the Wiener index W(G) , the transmission degrees and the parameter α∈[0,1] , and characterize the extremal graphs attaining these bounds

Extremal graphs for Estrada indices

Let $\mathcal{G}$ be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph $\mathcal{G}$ is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of $\mathcal{G}$, respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the family of graphs whose color classes have order not less than a fixed positive integer. The graphs for which the upper bound is tight is obtained. Additionally, an upper bound for the Estrada Index of the complement of a graph in the previous family of graphs with two color classes is given. A Nordhaus-Gaddum type inequality for the Laplacian Estrada index when {$\mathcal{G}$ is a bipartite} graph with color classes of order not less than $2$, is presented. Moreover, a sharp upper bound for the Estrada index of the line graph and for the signless Laplacian index of a graph in terms of connectivity is obtained.publishe

The spread of generalized reciprocal distance matrix

The generalized reciprocal distance matrix $RD_{\alpha}(G)$ was defined as $RD_{\alpha}(G)=\alpha RT(G)+(1-\alpha)RD(G),\quad 0\leq \alpha \leq 1.$ Let $\lambda_{1}(RD_{\alpha}(G))\geq \lambda_{2}(RD_{\alpha}(G))\geq \cdots \geq \lambda_{n}(RD_{\alpha}(G))$ be the eigenvalues of $RD_{\alpha}$ matrix of graphs $G$. Then the $RD_{\alpha}$-spread of graph $G$ can be defined as $S_{RD_{\alpha}}(G)=\lambda_{1}(RD_{\alpha}(G))-\lambda_{n}(RD_{\alpha}(G))$. In this paper, we first obtain some sharp lower and upper bounds for the $RD_{\alpha}$-spread of graphs. Then we determine the lower bounds for the $RD_{\alpha}$-spread of bipartite graphs and graphs with given clique number. At last, we give the $RD_{\alpha}$-spread of double star graphs. Our results generalize the related results of the reciprocal distance matrix and reciprocal distance signless Laplacian matrix.Comment: 14 page

Eigenvalue bounds for the signless laplacian

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures