Gaussianization of the spectra of graphs and networks. Theory and applications

Alhomaidhi, Alhanouf; Al-Thukair, Fawzi; Estrada, Ernesto

doi:10.1016/j.jmaa.2018.10.039

Cited by 6 publications

(14 citation statements)

References 45 publications

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“…Moreover, it is also related to the time-dependent Schrödinger equation with the squared Hamiltonian. Based on numerical simulations, H is found to be effective in differentiating the dynamics of particle hopping among bipartite and non-bipartite structures [24]. More results can be found in [28].…”

Section: Gaussian Estrada Index H Is Able To Describe the Partition Fmentioning

confidence: 97%

“…A different way to study graph spectra consists of analyzing matrix functions of the matrices associated with a graph or network. In analyzing graph invariants such as centrality and communicability, matrix functions of the form f (A) = ∑ ∞ k=0 c k A k have been found as a power tool [24]. When the gap λ 1 − λ 2 is large, EE tends to be dominated by the largest eigenvalue λ 1 .…”

Section: Motivationmentioning

confidence: 99%

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On Generalized Distance Gaussian Estrada Index of Graphs

Alhevaz

Shang

2019

Symmetry

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For a simple undirected connected graph G of order n, let D ( G ) , D L ( G ) , D Q ( G ) and T r ( 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 D α ( G ) is signified by D α ( G ) = α T r ( 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 P α ( G ) , as P α ( G ) = ∑ i = 1 n e - ∂ i 2 . Since characterization of P α ( G ) is very appealing in quantum information theory, it is interesting to study the quantity P α ( 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 P α ( 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.

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Section: Gaussian Estrada Index H Is Able To Describe the Partition Fmentioning

confidence: 97%

Section: Motivationmentioning

confidence: 99%

On Generalized Distance Gaussian Estrada Index of Graphs

Alhevaz

Shang

2019

Symmetry

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“…Because Lars Onsager (1903Onsager ( -1976 was the scientist who first study the negative absolute temperatures in [178] we propose to name the following index in his honor. 5 Definition 19 The Onsager Estrada index of G is defined as…”

Section: Negative Absolute Temperatures and The Onsager Estrada Indexmentioning

confidence: 99%

“…In this section we investigate Estrada indices that give higher weights to the contribution of eigenvalues other than the spectral radius. In particular we use here a technique known as spectral folding [36,229] to produce Gaussian Estrada indices [5,80]. In the following let λ be a given reference eigenvalue, I (z) be the modified Bessel function of the first kinds, Fig.…”

Section: Gaussian Estrada Indicesmentioning

confidence: 99%

The many facets of the Estrada indices of graphs and networks

Estrada

2021

SeMA

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The Estrada index of a graph/network is defined as the trace of the adjacency matrix exponential. It has been extended to other graph-theoretic matrices, such as the Laplacian, distance, Seidel adjacency, Harary, etc. Here, we describe many of these extensions, including new ones, such as Gaussian, Mittag–Leffler and Onsager ones. More importantly, we contextualize all of these indices in physico-mathematical frameworks which allow their interpretations and facilitate their extensions and further studies. We also describe several of the bounds and estimations of these indices reported in the literature and analyze many of them computationally for small graphs as well as large complex networks. This article is intended to formalize many of the Estrada indices proposed and studied in the mathematical literature serving as a guide for their further studies.

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“…These results are noteworthy in the sense that they not only contribute to the understanding of spectral theory of random networks but also presenting estimates to EE and LEE for almost all graphs (as the number of vertices goes to infinity), which are typically much sharper than previous bounds for fixed graphs. Important ramifications for distance Estrada index [18] and Gaussian Estrada index [19,20] have also been explored lately.…”

Section: Introductionmentioning

confidence: 99%

Estrada Index and Laplacian Estrada Index of Random Interdependent Graphs

Shang

2020

Mathematics

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Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by E E ( G ) = ∑ i = 1 n e λ i ( A ( G ) ) and L E E ( G ) = ∑ i = 1 n e λ i ( L ( G ) ) , where { λ i ( A ( G ) ) } i = 1 n and { λ i ( L ( G ) ) } i = 1 n are the eigenvalues of its adjacency and Laplacian matrices, respectively. In this paper, we establish almost sure upper bounds and lower bounds for random interdependent graph model, which is fairly general encompassing Erdös-Rényi random graph, random multipartite graph, and even stochastic block model. Our results unravel the non-triviality of interdependent edges between different constituting subgraphs in spectral property of interdependent graphs.

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Gaussianization of the spectra of graphs and networks. Theory and applications

Cited by 6 publications

References 45 publications

On Generalized Distance Gaussian Estrada Index of Graphs

On Generalized Distance Gaussian Estrada Index of Graphs

The many facets of the Estrada indices of graphs and networks

Estrada Index and Laplacian Estrada Index of Random Interdependent Graphs

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