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12 pages, 1689 KiB

Open AccessArticle

Wiener–Hosoya Matrix of Connected Graphs

by Hassan Ibrahim, Reza Sharafdini, Tamás Réti and Abolape Akwu

Mathematics 2021, 9(4), 359; https://0-doi-org.brum.beds.ac.uk/10.3390/math9040359 - 11 Feb 2021

Cited by 2 | Viewed by 1687

Abstract

Let G be a connected (molecular) graph with the vertex set

V (G) = {v_{1}, \dots, v_{n}}

, and let

d_{i}

and

σ_{i}

denote, respectively, the vertex degree and the transmission of [...] Read more.

Let G be a connected (molecular) graph with the vertex set

V (G) = {v_{1}, \dots, v_{n}}

, and let

d_{i}

and

σ_{i}

denote, respectively, the vertex degree and the transmission of

v_{i}

, for

1 \leq i \leq n

. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the

n \times n

matrix whose

(i, j)

-entry is equal to

\frac{σ_{i}}{2 d_{i}} + \frac{σ_{j}}{2 d_{j}}

if

v_{i}

and

v_{j}

are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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18 pages, 3539 KiB

Open AccessArticle

Skeletal Rearrangements of the C₂₄₀ Fullerene: Efficient Topological Descriptors for Monitoring Stone–Wales Transformations

by Denis Sh. Sabirov and Ottorino Ori

Mathematics 2020, 8(6), 968; https://0-doi-org.brum.beds.ac.uk/10.3390/math8060968 - 12 Jun 2020

Cited by 17 | Viewed by 3299

Abstract

Stone–Wales rearrangements of the fullerene surface are an uncharted field in theoretical chemistry. Here, we study them on the example of the giant icosahedral fullerene C₂₄₀ to demonstrate the complex chemical mechanisms emerging on its carbon skeleton. The Stone–Wales transformations of C [...] Read more.

Stone–Wales rearrangements of the fullerene surface are an uncharted field in theoretical chemistry. Here, we study them on the example of the giant icosahedral fullerene C₂₄₀ to demonstrate the complex chemical mechanisms emerging on its carbon skeleton. The Stone–Wales transformations of C₂₄₀ can produce the defected isomers containing heptagons, extra pentagons and other unordinary rings. Their formations have been described in terms of (i) quantum-chemically calculated energetic, molecular, and geometric parameters; and (ii) topological indices. We have found the correlations between the quantities from the two sets that point out the role of long-range topological defects in governing the formation and the chemical reactivity of fullerene molecules. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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9 pages, 298 KiB

Open AccessArticle

On the A_α-Spectral Radii of Cactus Graphs

by Chunxiang Wang, Shaohui Wang, Jia-Bao Liu and Bing Wei

Mathematics 2020, 8(6), 869; https://0-doi-org.brum.beds.ac.uk/10.3390/math8060869 - 28 May 2020

Cited by 2 | Viewed by 1967

Abstract

Let

A (G)

be the adjacent matrix and

D (G)

the diagonal matrix of the degrees of a graph G, respectively. For

0 \leq α \leq 1

, the

A_{α}

-matrix is the general adjacency and signless [...] Read more.

Let

A (G)

be the adjacent matrix and

D (G)

the diagonal matrix of the degrees of a graph G, respectively. For

0 \leq α \leq 1

, the

A_{α}

-matrix is the general adjacency and signless Laplacian spectral matrix having the form of

A_{α} (G) = α D (G) + (1 - α) A (G)

. Clearly,

A_{0} (G)

is the adjacent matrix and

2 A_{\frac{1}{2}}

is the signless Laplacian matrix. A cactus is a connected graph such that any two of its cycles have at most one common vertex, that is an extension of the tree. The

A_{α}

-spectral radius of a cactus graph with n vertices and k cycles is explored. The outcomes obtained in this paper can imply some previous bounds from trees to cacti. In addition, the corresponding extremal graphs are determined. Furthermore, we proposed all eigenvalues of such extremal cacti. Our results extended and enriched previous known results. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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13 pages, 309 KiB

Open AccessArticle

Wiener Index of Edge Thorny Graphs of Catacondensed Benzenoids

by Andrey A. Dobrynin and Ali Iranmanesh

Mathematics 2020, 8(4), 467; https://0-doi-org.brum.beds.ac.uk/10.3390/math8040467 - 27 Mar 2020

Cited by 4 | Viewed by 8152

Abstract

The Wiener index is a topological index of a molecular graph, defined as the sum of distances between all pairs of its vertices. Benzenoid graphs include molecular graphs of polycyclic aromatic hydrocarbons. An edge thorny graph G is constructed from a catacondensed benzenoid [...] Read more.

The Wiener index is a topological index of a molecular graph, defined as the sum of distances between all pairs of its vertices. Benzenoid graphs include molecular graphs of polycyclic aromatic hydrocarbons. An edge thorny graph G is constructed from a catacondensed benzenoid graph H by attaching new graphs to edges of a perfect matching of H. A formula for the Wiener index of G is derived. The index of the resulting graph does not contain distance characteristics of elements of H and depends on the Wiener index of H and distance properties of the attached graphs. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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13 pages, 1075 KiB

Open AccessArticle

Novel Face Index for Benzenoid Hydrocarbons

by Muhammad Kamran Jamil, Muhammad Imran and Kanza Abdul Sattar

Mathematics 2020, 8(3), 312; https://0-doi-org.brum.beds.ac.uk/10.3390/math8030312 - 1 Mar 2020

Cited by 20 | Viewed by 2971

Abstract

A novel topological index, the face index (

F I

), is proposed in this paper. For a molecular graph G, face index is defined as [...] Read more.

A novel topological index, the face index (

F I

), is proposed in this paper. For a molecular graph G, face index is defined as

F I (G) = \sum_{f \in F (G)} d (f) = \sum_{v \sim f, f \in F (G)} d (v)

, where

d (v)

is the degree of the vertex v. The index is very easy to calculate and improved the previously discussed correlation models for

π - e l e c t r o n

energy and boiling point of benzenoid hydrocarbons. The study shows that the multiple linear regression involving the novel topological index can predict the

π

-electron energy and boiling points of the benzenoid hydrocarbons with correlation coefficient

r > 0.99

. Moreover, the face indices of some planar molecular structures such as 2-dimensional graphene, triangular benzenoid, circumcoronene series of benzenoid are also investigated. The results suggest that the proposed index with good correlation ability and structural selectivity promised to be a useful parameter in QSPR/QSAR. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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12 pages, 313 KiB

Open AccessArticle

Some Bounds on Zeroth-Order General Randić Index

by Muhammad Kamran Jamil, Ioan Tomescu, Muhammad Imran and Aisha Javed

Mathematics 2020, 8(1), 98; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010098 - 7 Jan 2020

Cited by 5 | Viewed by 2494

Abstract

For a graph G without isolated vertices, the inverse degree of a graph G is defined as

I D (G) = \sum_{u \in V (G)} d {(u)}^{- 1}

where

d (u)

is the [...] Read more.

For a graph G without isolated vertices, the inverse degree of a graph G is defined as

I D (G) = \sum_{u \in V (G)} d {(u)}^{- 1}

where

d (u)

is the number of vertices adjacent to the vertex u in G. By replacing

- 1

by any non-zero real number we obtain zeroth-order general Randić index, i.e.,

^{0} R_{γ} (G) = \sum_{u \in V (G)} d {(u)}^{γ}

, where

γ \in R - {0}

. Xu et al. investigated some lower and upper bounds on

I D

for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for

γ < 0

. The corresponding extremal graphs have also been identified. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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12 pages, 302 KiB

Open AccessArticle

Enumeration of Pentahexagonal Annuli in the Plane

by Andrey A. Dobrynin and Vladimir R. Rosenfeld

Mathematics 2019, 7(12), 1156; https://0-doi-org.brum.beds.ac.uk/10.3390/math7121156 - 1 Dec 2019

Viewed by 2357

Abstract

Pentahexagonal annuli are closed chains consisting of regular pentagons and hexagons. Such configurations can be easily recognized in various complex designs, in particular, in molecular carbon constructions. Results of computer enumeration of annuli without overlapping on the plane are presented for up to [...] Read more.

Pentahexagonal annuli are closed chains consisting of regular pentagons and hexagons. Such configurations can be easily recognized in various complex designs, in particular, in molecular carbon constructions. Results of computer enumeration of annuli without overlapping on the plane are presented for up to 18 pentagons and hexagons. We determine how many annuli have certain properties for a fixed number of pentagons. In particular, we consider symmetry, pentagon separation (the least ring-distance between pentagons), uniformity of pentagon distribution, and pentagonal thickness (the size of maximal connected part of pentagons) of annuli. Pictures of all annuli with the number of pentagons and hexagons up to 17 are presented (more than 1300 diagrams). Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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12 pages, 1698 KiB

Open AccessArticle

Total and Double Total Domination Number on Hexagonal Grid

by Antoaneta Klobučar and Ana Klobučar

Mathematics 2019, 7(11), 1110; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111110 - 15 Nov 2019

Cited by 6 | Viewed by 2718

Abstract

In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid

H_{m, n}

with m hexagons in a row and n hexagons [...] Read more.

In this paper, we determine the upper and lower bound for the total domination number and exact values and the upper bound for the double-total domination number on hexagonal grid

H_{m, n}

with m hexagons in a row and n hexagons in a column. Further, we explore the ratio between the total domination number and the number of vertices of

H_{m, n}

when m and n tend to infinity. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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17 pages, 397 KiB

Open AccessArticle

On the Wiener Complexity and the Wiener Index of Fullerene Graphs

by Andrey A. Dobrynin and Andrei Yu Vesnin

Mathematics 2019, 7(11), 1071; https://0-doi-org.brum.beds.ac.uk/10.3390/math7111071 - 7 Nov 2019

Cited by 11 | Viewed by 2819

Abstract

Fullerenes are molecules that can be presented in the form of cage-like polyhedra, consisting only of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is a local graph invariant defined as the [...] Read more.

Fullerenes are molecules that can be presented in the form of cage-like polyhedra, consisting only of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex v of a graph is a local graph invariant defined as the sum of distances from v to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order

n \leq 232

and IPR fullerene graphs of order

n \leq 270

are presented. The structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed, and formulas for the Wiener index of several families of graphs are obtained. Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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Review

Jump to: Research

23 pages, 5548 KiB

Open AccessFeature PaperReview

Properties of Entropy-Based Topological Measures of Fullerenes

by Modjtaba Ghorbani, Matthias Dehmer and Frank Emmert-Streib

Mathematics 2020, 8(5), 740; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050740 - 7 May 2020

Cited by 10 | Viewed by 2755

Abstract

A fullerene is a cubic three-connected graph whose faces are entirely composed of pentagons and hexagons. Entropy applied to graphs is one of the significant approaches to measuring the complexity of relational structures. Recently, the research on complex networks has received great attention, [...] Read more.

A fullerene is a cubic three-connected graph whose faces are entirely composed of pentagons and hexagons. Entropy applied to graphs is one of the significant approaches to measuring the complexity of relational structures. Recently, the research on complex networks has received great attention, because many complex systems can be modelled as networks consisting of components as well as relations among these components. Information—theoretic measures have been used to analyze chemical structures possessing bond types and hetero-atoms. In the present article, we reviewed various entropy-based measures on fullerene graphs. In particular, we surveyed results on the topological information content of a graph, namely the orbit-entropy I_a(G), the symmetry index, a degree-based entropy measure I_λ(G), the eccentric-entropy If_σ(G) and the Hosoya entropy H(G). Full article

(This article belongs to the Special Issue Graph Theory at Work in Carbon Chemistry)

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