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13 pages, 413 KB

Open AccessArticle

Fractional-Modified Bessel Function of the First Kind of Integer Order

by Andrés Martín and Ernesto Estrada

Mathematics 2023, 11(7), 1630; https://doi.org/10.3390/math11071630 - 28 Mar 2023

Cited by 2 | Viewed by 3797

The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the [...] Read more.

The modified Bessel function (MBF) of the first kind is a fundamental special function in mathematics with applications in a large number of areas. When the order of this function is integer, it has an integral representation which includes the exponential of the cosine function. Here, we generalize this MBF to include a fractional parameter, such that the exponential in the previously mentioned integral is replaced by a Mittag–Leffler function. The necessity for this generalization arises from a problem of communication in networks. We find the power series representation of the fractional MBF of the first kind as well as some differential properties. We give some examples of its utility in graph/networks analysis and mention some fundamental open problems for further investigation. Full article

(This article belongs to the Special Issue Mathematics: 10th Anniversary)

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14 pages, 314 KB

Open AccessArticle

Generalized Randić Estrada Indices of Graphs

by Eber Lenes, Exequiel Mallea-Zepeda, Luis Medina and Jonnathan Rodríguez

Mathematics 2022, 10(16), 2932; https://doi.org/10.3390/math10162932 - 14 Aug 2022

Viewed by 1823

Abstract

Let G be a simple undirected graph on n vertices. V. Nikiforov studied hybrids of

A_{G}

and

D_{G}

and defined the matrix

A_{α}^{G}

for every real

α \in [0, 1]

as [...] Read more.

Let G be a simple undirected graph on n vertices. V. Nikiforov studied hybrids of

A_{G}

and

D_{G}

and defined the matrix

A_{α}^{G}

for every real

α \in [0, 1]

as

A_{α}^{G} = α D_{G} + (1 - α) A_{G} .

In this paper, we define the generalized Randić matrix for graph G, and we introduce and establish bounds for the Estrada index of this new matrix. Furthermore, we find the smallest value of

α

for which the generalized Randić matrix is positive semidefinite. Finally, we present the solution to the problem proposed by V. Nikiforov. The problem consists of the following: for a given simple undirected graph G, determine the smallest value of

α

for which

A_{α}^{G}

is positive semidefinite. Full article

(This article belongs to the Special Issue Advances in Discrete Applied Mathematics and Graph Theory, 2nd Edition)

11 pages, 413 KB

Open AccessArticle

The Extremal Structures of r-Uniform Unicyclic Hypergraphs on the Signless Laplacian Estrada Index

by Hongyan Lu and Zhongxun Zhu

Mathematics 2022, 10(6), 941; https://doi.org/10.3390/math10060941 - 15 Mar 2022

Cited by 1 | Viewed by 1611

Abstract

S L E E

has various applications in a large variety of problems. The signless Laplacian Estrada index of a hypergraph H is defined as [...] Read more.

S L E E

has various applications in a large variety of problems. The signless Laplacian Estrada index of a hypergraph H is defined as

S L E E (H) = \sum_{i = 1}^{n} e^{λ_{i} (Q)}

, where

λ_{1} (Q), λ_{2} (Q), \dots, λ_{n} (Q)

are the eigenvalues of the signless Laplacian matrix of H. In this paper, we characterize the unique r-uniform unicyclic hypergraphs with maximum and minimum

S L E E

. Full article

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11 pages, 357 KB

Open AccessArticle

On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

by Wenjie Ning and Kun Wang

Mathematics 2021, 9(19), 2395; https://doi.org/10.3390/math9192395 - 26 Sep 2021

Cited by 1 | Viewed by 2030

Abstract

The Estrada index of a graph G is defined as

E E (G) = \sum_{i = 1}^{n} e^{λ_{i}}

, where

λ_{1}, λ_{2}, \dots, λ_{n}

are the eigenvalues of the adjacency matrix [...] Read more.

The Estrada index of a graph G is defined as

E E (G) = \sum_{i = 1}^{n} e^{λ_{i}}

, where

λ_{1}, λ_{2}, \dots, λ_{n}

are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let

U (n, d)

be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among

U (n, d)

. We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even. Full article

(This article belongs to the Special Issue The Matrix Theory of Graphs)

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11 pages, 579 KB

Open AccessArticle

Effect of a Ring onto Values of Eigenvalue–Based Molecular Descriptors

by Izudin Redžepović, Slavko Radenković and Boris Furtula

Symmetry 2021, 13(8), 1515; https://doi.org/10.3390/sym13081515 - 18 Aug 2021

Cited by 5 | Viewed by 2297

Abstract

The eigenvalues of the characteristic polynomial of a graph are sensitive to its symmetry-related characteristics. Within this study, we have examined three eigenvalue–based molecular descriptors. These topological molecular descriptors, among others, are gathering information on the symmetry of a molecular graph. Furthermore, they [...] Read more.

The eigenvalues of the characteristic polynomial of a graph are sensitive to its symmetry-related characteristics. Within this study, we have examined three eigenvalue–based molecular descriptors. These topological molecular descriptors, among others, are gathering information on the symmetry of a molecular graph. Furthermore, they are being ordinarily employed for predicting physico–chemical properties and/or biological activities of molecules. It has been shown that these indices describe well molecular features that are depending on fine structural details. Therefore, revealing the impact of structural details on the values of the eigenvalue–based topological indices should give a hunch how physico–chemical properties depend on them as well. Here, an effect of a ring in a molecule on the values of the graph energy, Estrada index and the resolvent energy of a graph is examined. Full article

(This article belongs to the Special Issue Discrete and Fractional Mathematics: Symmetry and Applications)

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7 pages, 248 KB

Open AccessArticle

A Note on the Estrada Index of the A_α-Matrix

by Jonnathan Rodríguez and Hans Nina

Mathematics 2021, 9(8), 811; https://doi.org/10.3390/math9080811 - 8 Apr 2021

Cited by 1 | Viewed by 1802

Abstract

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of

A (G)

and

D (G)

[...] Read more.

Let G be a graph on n vertices. The Estrada index of G is an invariant that is calculated from the eigenvalues of the adjacency matrix of a graph. V. Nikiforov studied hybrids of

A (G)

and

D (G)

and defined the

A_{α}

-matrix for every real

α \in [0, 1]

as:

A_{α} (G) = α D (G) + (1 - α) A (G) .

In this paper, using a different demonstration technique, we present a way to compare the Estrada index of the

A_{α}

-matrix with the Estrada index of the adjacency matrix of the graph G. Furthermore, lower bounds for the Estrada index are established. Full article

8 pages, 256 KB

Open AccessArticle

Estrada Index and Laplacian Estrada Index of Random Interdependent Graphs

by Yilun Shang

Mathematics 2020, 8(7), 1063; https://doi.org/10.3390/math8071063 - 1 Jul 2020

Cited by 10 | Viewed by 3044

Abstract

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by

E E (G) = \sum_{i = 1}^{n} e^{λ_{i} (A (G))}

and [...] Read more.

Let G be a simple graph of order n. The Estrada index and Laplacian Estrada index of G are defined by

E E (G) = \sum_{i = 1}^{n} e^{λ_{i} (A (G))}

and

L E E (G) = \sum_{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. Full article

(This article belongs to the Special Issue Algebra and Its Applications)

24 pages, 349 KB

Open AccessArticle

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

by Abdollah Alhevaz, Maryam Baghipur and Yilun Shang

Mathematics 2019, 7(10), 995; https://doi.org/10.3390/math7100995 - 19 Oct 2019

Cited by 17 | Viewed by 2638

Abstract

Suppose that G is a simple undirected connected graph. Denote by

D (G)

the distance matrix of G and by

T r (G)

the diagonal matrix of the vertex transmissions in G, and let [...] Read more.

Suppose that G is a simple undirected connected graph. Denote by

D (G)

the distance matrix of G and by

T r (G)

the diagonal matrix of the vertex transmissions in G, and let

α \in [0, 1]

. The generalized distance matrix

D_{α} (G)

is defined as

D_{α} (G) = α T r (G) + (1 - α) D (G)

, where

0 \leq α \leq 1

. If

\partial_{1} \geq \partial_{2} \geq \dots \geq \partial_{n}

are the eigenvalues of

D_{α} (G)

; we define the generalized distance Estrada index of the graph G as

D_{α} E (G) = \sum_{i = 1}^{n} e^{(\partial_{i} - \frac{2 α W (G)}{n})},

where

W (G)

denotes for the Wiener index of G. It is clear from the definition that

D_{0} E (G) = D E E (G)

and

2 D_{\frac{1}{2}} E (G) = D^{Q} E E (G)

, where

D E E (G)

denotes the distance Estrada index of G and

D^{Q} E E (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

D_{α} E (G)

of some special classes of graphs. Full article

(This article belongs to the Section E1: Mathematics and Computer Science)

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21 pages, 364 KB

Open AccessArticle

On Generalized Distance Gaussian Estrada Index of Graphs

by Abdollah Alhevaz, Maryam Baghipur and Yilun Shang

Symmetry 2019, 11(10), 1276; https://doi.org/10.3390/sym11101276 - 11 Oct 2019

Cited by 12 | Viewed by 3070

Abstract

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, [...] Read more.

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 \in [0, 1] .

Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let

\partial_{1}, \partial_{2}, \dots, \partial_{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) = \sum_{i = 1}^{n} e^{- \partial_{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 \in [0, 1]

, and characterize the extremal graphs attaining these bounds. Full article

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6 pages, 239 KB

Open AccessArticle

Lower Bounds for Gaussian Estrada Index of Graphs

by Yilun Shang

Symmetry 2018, 10(8), 325; https://doi.org/10.3390/sym10080325 - 7 Aug 2018

Cited by 20 | Viewed by 3436

Abstract

Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by

λ_{1}, λ_{2}, \dots, λ_{n}

. The Gaussian Estrada index, denoted by

H (G)

(Estrada et al., Chaos [...] Read more.

Suppose that G is a graph over n vertices. G has n eigenvalues (of adjacency matrix) represented by

λ_{1}, λ_{2}, \dots, λ_{n}

. The Gaussian Estrada index, denoted by

H (G)

(Estrada et al., Chaos 27(2017) 023109), can be defined as

H (G) = \sum_{i = 1}^{n} e^{- λ_{i}^{2}}

. Gaussian Estrada index underlines the eigenvalues close to zero, which plays an important role in chemistry reactions, such as molecular stability and molecular magnetic properties. In a network of particles governed by quantum mechanics, this graph-theoretic index is known to account for the information encoded in the eigenvalues of the Hamiltonian near zero by folding the graph spectrum. In this paper, we establish some new lower bounds for

H (G)

in terms of the number of vertices, the number of edges, as well as the first Zagreb index. Full article

11 pages, 237 KB

Open AccessArticle

Estrada Index of Random Bipartite Graphs

by Yilun Shang

Symmetry 2015, 7(4), 2195-2205; https://doi.org/10.3390/sym7042195 - 7 Dec 2015

Cited by 8 | Viewed by 7013

Abstract

The Estrada index of a graph \(G\) of \(n\) vertices is defined by \(EE(G)=\sum_{i=1}^ne^{\lambda_i}\), where \(\lambda_1,\lambda_2,\cdots,\lambda_n\) are the eigenvalues of \(G\). In this paper, we give upper and lower bounds of \(EE(G)\) for almost all bipartite graphs by investigating the upper and lower [...] Read more.

The Estrada index of a graph \(G\) of \(n\) vertices is defined by \(EE(G)=\sum_{i=1}^ne^{\lambda_i}\), where \(\lambda_1,\lambda_2,\cdots,\lambda_n\) are the eigenvalues of \(G\). In this paper, we give upper and lower bounds of \(EE(G)\) for almost all bipartite graphs by investigating the upper and lower bounds of the spectrum of random matrices. We also formulate an exact estimate of \(EE(G)\) for almost all balanced bipartite graphs. Full article

8 pages, 202 KB

Open AccessTechnical Note

Estrada and L-Estrada Indices of Edge-Independent Random Graphs

by Yilun Shang

Symmetry 2015, 7(3), 1455-1462; https://doi.org/10.3390/sym7031455 - 19 Aug 2015

Cited by 16 | Viewed by 4489

Abstract

Let \(G\) be a simple graph of order \(n\) with eigenvalues \(\lambda_1,\lambda_2,\cdots,\lambda_n\) and normalized Laplacian eigenvalues \(\mu_1,\mu_2,\cdots,\mu_n\). The Estrada index and normalized Laplacian Estrada index are defined as \(EE(G)=\sum_{k=1}^ne^{\lambda_k}\) and \(\mathcal{L}EE(G)=\sum_{k=1}^ne^{\mu_k-1}\), respectively. We establish upper and lower bounds to \(EE\) and \(\mathcal{L}EE\) for [...] Read more.

Let \(G\) be a simple graph of order \(n\) with eigenvalues \(\lambda_1,\lambda_2,\cdots,\lambda_n\) and normalized Laplacian eigenvalues \(\mu_1,\mu_2,\cdots,\mu_n\). The Estrada index and normalized Laplacian Estrada index are defined as \(EE(G)=\sum_{k=1}^ne^{\lambda_k}\) and \(\mathcal{L}EE(G)=\sum_{k=1}^ne^{\mu_k-1}\), respectively. We establish upper and lower bounds to \(EE\) and \(\mathcal{L}EE\) for edge-independent random graphs, containing the classical Erdös-Rényi graphs as special cases. Full article

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