Sharp Bounds for the Signless Laplacian Spectral Radius in Terms of Clique Number

In this paper, we present a sharp upper and lower bounds for the signless Laplacian spectral radius of graphs in terms of clique number. Moreover, the extremal graphs which attain the upper and lower bounds are characterized. In addition, these results disprove the two conjectures on the signless Laplacian spectral radius in [P. Hansen and C. Lucas, Bounds and conjectures for the signless Laplacian index of graphs, Linear Algebra Appl., 432(2010) 3319-3336].Comment: 15 pages 1 figure; linear algebra and its applications 201

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

Sharp Bounds on (Generalized) Distance Energy of Graphs

Given a simple connected graph G, let D(G) be the distance matrix, DL(G) be the distance Laplacian matrix, DQ(G) be the distance signless Laplacian matrix, and Tr(G) be the vertex transmission diagonal matrix of G. We introduce the generalized distance matrix Dα(G)=αTr(G)+(1−α)D(G) , where α∈[0,1] . Noting that D0(G)=D(G),2D12(G)=DQ(G),D1(G)=Tr(G) and Dα(G)−Dβ(G)=(α−β)DL(G) , we reveal that a generalized distance matrix ideally bridges the spectral theories of the three constituent matrices. In this paper, we obtain some sharp upper and lower bounds for the generalized distance energy of a graph G involving different graph invariants. As an application of our results, we will be able to improve some of the recently given bounds in the literature for distance energy and distance signless Laplacian energy of graphs. The extremal graphs of the corresponding bounds are also characterized

Merging the A- and Q-spectral theories

Let $G$ be a graph with adjacency matrix $A\left( G\right)$, and let $D\left( G\right)$ be the diagonal matrix of the degrees of $G.$ The signless Laplacian $Q\left( G\right)$ of $G$ is defined as $Q\left( G\right) :=A\left( G\right) +D\left( G\right)$. Cvetkovi\'{c} called the study of the adjacency matrix the $A$% \textit{-spectral theory}, and the study of the signless Laplacian--the $Q$\textit{-spectral theory}. During the years many similarities and differences between these two theories have been established. To track the gradual change of $A\left( G\right)$ into $Q\left( G\right)$ in this paper it is suggested to study the convex linear combinations $A_{\alpha }\left( G\right)$ of $A\left( G\right)$ and $D\left( G\right)$ defined by $$A_{\alpha}\left( G\right) :=\alpha D\left( G\right) +\left( 1-\alpha\right) A\left( G\right) \text{, \ \ }0\leq\alpha\leq1.$$ This study sheds new light on $A\left( G\right)$ and $Q\left( G\right)$, and yields some surprises, in particular, a novel spectral Tur\'{a}n theorem. A number of challenging open problems are discussed.Comment: 26 page

A Sharp upper bound for the spectral radius of a nonnegative matrix and applications

In this paper, we obtain a sharp upper bound for the spectral radius of a nonnegative matrix. This result is used to present upper bounds for the adjacency spectral radius, the Laplacian spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance Laplacian spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph. These results are new or generalize some known results.Comment: 16 pages in Czechoslovak Math. J., 2016. arXiv admin note: text overlap with arXiv:1507.0705