Merging the Spectral Theories of Distance Estrada and Distance Signless Laplacian Estrada Indices of Graphs

Suppose that G is a simple undirected connected graph. Denote by D(G) the distance matrix of G and by Tr(G) the diagonal matrix of the vertex transmissions in G, and let α∈[0,1] . The generalized distance matrix Dα(G) is defined as Dα(G)=αTr(G)+(1−α)D(G) , where 0≤α≤1 . If ∂1≥∂2≥…≥∂n are the eigenvalues of Dα(G) ; we define the generalized distance Estrada index of the graph G as DαE(G)=∑ni=1e(∂i−2αW(G)n), where W(G) denotes for the Wiener index of G. It is clear from the definition that D0E(G)=DEE(G) and 2D12E(G)=DQEE(G) , where DEE(G) denotes the distance Estrada index of G and DQEE(G) denotes the distance signless Laplacian Estrada index of G. This shows that the concept of generalized distance Estrada index of a graph G merges the theories of distance Estrada index and the distance signless Laplacian Estrada index. In this paper, we obtain some lower and upper bounds for the generalized distance Estrada index, in terms of various graph parameters associated with the structure of the graph G, and characterize the extremal graphs attaining these bounds. We also highlight relationship between the generalized distance Estrada index and the other graph-spectrum-based invariants, including generalized distance energy. Moreover, we have worked out some expressions for DαE(G) of some special classes of graphs

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

Bounds on the Distance Energy and the Distance Estrada Index of Strongly Quotient Graphs

The notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada index of CSQG whose diameter does not exceed two. Additionally, we show that our results improve most of the results obtained by Güngör and Bozkurt (2009) and Zaferani (2008)

A majorization method for localizing graph topological indices

This paper presents a unified approach for localizing some relevant graph topological indices via majorization techniques. Through this method, old and new bounds are derived and numerical examples are provided, showing how former results in the literature could be improved.Comment: 11 page

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