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13 pages, 480 KiB

Open AccessArticle

On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices

by Abeer M. Albalahi, Muhammad Rizwan, Akhlaq A. Bhatti, Ivan Gutman, Akbar Ali, Tariq Alraqad and Hicham Saber

Axioms 2025, 14(1), 23; https://doi.org/10.3390/axioms14010023 - 30 Dec 2024

Viewed by 675

Abstract

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index

S E I_{a}

and the general zeroth-order Randić index

{}^{0}R_{α}

. The minimum values of

S E I_{a}

[...] Read more.

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index

S E I_{a}

and the general zeroth-order Randić index

{}^{0}R_{α}

. The minimum values of

S E I_{a}

and

{}^{0}R_{α}

in the class of all trees of fixed order containing no vertex of even degree are obtained for

a > 1

and

α \in [0, 1]

; also, the maximum value of

{}^{0}R_{α}

in the mentioned class is determined for

0 < α < 1

. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of

S E I_{a}

and

{}^{0}R_{α}

are found for every real number

α \notin {0, 1}

and

a > 1

. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of

S E I_{a}

and

{}^{0}R_{α}

are determined when

a > 1

and

α

does not belong to the closed interval

[0, 1]

; in the same class, the maximum values of

{}^{0}R_{α}

are also found for

0 < α < 1

. The graphs that achieve the obtained extremal values are also determined. Full article

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12 pages, 412 KiB

Open AccessArticle

General Randić Index of Unicyclic Graphs and Its Applications to Drugs

by Alaa Altassan and Muhammad Imran

Symmetry 2024, 16(1), 113; https://doi.org/10.3390/sym16010113 - 18 Jan 2024

Cited by 2 | Viewed by 1867

Abstract

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for

η_{0} \leq η < 0

among all n-vertex unicyclic graphs with a fixed maximum degree

Δ

and the maximum and the second maximum [...] Read more.

In this work, we determine the maximum general Randić index (a general symmetric function of vertex degrees) for

η_{0} \leq η < 0

among all n-vertex unicyclic graphs with a fixed maximum degree

Δ

and the maximum and the second maximum general Randić index for

η_{0} \leq η < 0

among all n-vertex unicyclic graphs, where

η_{0} \approx - 0.21

. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randić index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randić index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randić index with the physical properties of drugs on the said chemical trees. We show that the Randić index is better correlated with the heat of vaporization for these alkanes. Full article

(This article belongs to the Section Mathematics)

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16 pages, 321 KiB

Open AccessArticle

Entropy and Multi-Fractal Analysis in Complex Fractal Systems Using Graph Theory

by Zeeshan Saleem Mufti, Ali H. Tedjani, Rukhshanda Anjum and Turki Alsuraiheed

Axioms 2023, 12(12), 1126; https://doi.org/10.3390/axioms12121126 - 15 Dec 2023

Cited by 2 | Viewed by 1407

Abstract

In 1997, Sierpinski graphs,

S (n, k)

, were obtained by Klavzar and Milutinovic. The graph

S (1, k)

represents the complete graph

K_{k}

and

S (n, 3)

is known as the graph [...] Read more.

In 1997, Sierpinski graphs,

S (n, k)

, were obtained by Klavzar and Milutinovic. The graph

S (1, k)

represents the complete graph

K_{k}

and

S (n, 3)

is known as the graph of the Tower of Hanoi. Through generalizing the notion of a Sierpinski graph, a graph named a generalized Sierpinski graph, denoted by

S i e (Λ, t)

, already exists in the literature. For every graph, numerous polynomials are being studied, such as chromatic polynomials, matching polynomials, independence polynomials, and the M-polynomial. For every polynomial there is an underlying geometrical object which extracts everything that is hidden in a polynomial of a common framework. Now, we describe the steps by which we complete our task. In the first step, we generate an M-polynomial for a generalized Sierpinski graph

S i e (Λ, t)

. In the second step, we extract some degree-based indices of a generalized Sierpinski graph

S i e (Λ, t)

using the M-polynomial generated in step 1. In step 3, we generate the entropy of a generalized Sierpinski graph

S i e (Λ, t)

by using the Randić index. Full article

(This article belongs to the Special Issue Advances in Graph Theory and Combinatorial Optimization)

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11 pages, 839 KiB

Open AccessArticle

Generalized Quasi Trees with Respect to Degree Based Topological Indices and Their Applications to COVID-19 Drugs

by Alaa Altassan and Muhammad Imran

Mathematics 2023, 11(3), 647; https://doi.org/10.3390/math11030647 - 27 Jan 2023

Cited by 3 | Viewed by 1713

Abstract

The l-generalized quasi tree is a graph G for which we can find

W \subset V (G)

with

| W | = l

such that

G - W

is a tree but for an arbitrary [...] Read more.

The l-generalized quasi tree is a graph G for which we can find

W \subset V (G)

with

| W | = l

such that

G - W

is a tree but for an arbitrary

Y \subset V (G)

with

| Y | < l

,

G - Y

is not a tree. In this paper, inequalities with respect to zeroth-order Randić and hyper-Zagreb indices are studied in the class of l-generalized quasi trees. The corresponding extremal graphs corresponding to these indices in the class of l-generalized quasi trees are also obtained. In addition, we carry QSPR analysis of COVID-19 drugs with zeroth-order Randić and hyper-Zagreb indices (energy). Full article

(This article belongs to the Special Issue Graph Theory and Applications)

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14 pages, 314 KiB

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 1716

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)

16 pages, 355 KiB

Open AccessArticle

On the Topological Indices of Commuting Graphs for Finite Non-Abelian Groups

by Fawad Ali, Bilal A. Rather, Nahid Fatima, Muhammad Sarfraz, Asad Ullah, Khalid Abdulkhaliq M. Alharbi and Rahim Dad

Symmetry 2022, 14(6), 1266; https://doi.org/10.3390/sym14061266 - 19 Jun 2022

Cited by 20 | Viewed by 2720

Abstract

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It [...] Read more.

A topological index is a number generated from a molecular structure (i.e., a graph) that indicates the essential structural properties of the proposed molecule. Indeed, it is an algebraic quantity connected with the chemical structure that correlates it with various physical characteristics. It is possible to determine several different properties, such as chemical activity, thermodynamic properties, physicochemical activity, and biological activity, using several topological indices, such as the geometric-arithmetic index, arithmetic-geometric index, Randić index, and the atom-bond connectivity indices. Consider

G

as a group and H as a non-empty subset of

G

. The commuting graph

C (G, H)

, has H as the vertex set, where

h_{1}, h_{2} \in H

are edge connected whenever

h_{1}

and

h_{2}

commute in

G

. This article examines the topological characteristics of commuting graphs having an algebraic structure by computing their atomic-bond connectivity index, the Wiener index and its reciprocal, the harmonic index, geometric-arithmetic index, Randić index, Harary index, and the Schultz molecular topological index. Moreover, we study the Hosoya properties, such as the Hosoya polynomial and the reciprocal statuses of the Hosoya polynomial of the commuting graphs of finite subgroups of

SL (2, C)

. Finally, we compute the Z-index of the commuting graphs of the binary dihedral groups. Full article

(This article belongs to the Special Issue Topological Indices and Symmetry in Complex Networks)

17 pages, 275 KiB

Open AccessArticle

Inequalities on the Generalized ABC Index

by Paul Bosch, Edil D. Molina, José M. Rodríguez and José M. Sigarreta

Mathematics 2021, 9(10), 1151; https://doi.org/10.3390/math9101151 - 20 May 2021

Cited by 6 | Viewed by 2291

Abstract

In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for

A B C_{α}

improved, when

α = 1 / 2

, known results on the

A B C

index. [...] Read more.

In this work, we obtained new results relating the generalized atom-bond connectivity index with the general Randić index. Some of these inequalities for

A B C_{α}

improved, when

α = 1 / 2

, known results on the

A B C

index. Moreover, in order to obtain our results, we proved a kind of converse Hölder inequality, which is interesting on its own. Full article

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

24 pages, 433 KiB

Open AccessArticle

On Valency-Based Molecular Topological Descriptors of Subdivision Vertex-Edge Join of Three Graphs

by Juan L. G. Guirao, Muhammad Imran, Muhammad Kamran Siddiqui and Shehnaz Akhter

Symmetry 2020, 12(6), 1026; https://doi.org/10.3390/sym12061026 - 17 Jun 2020

Cited by 16 | Viewed by 2720

Abstract

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of [...] Read more.

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of the ease of generation and the speed with which these computations can be accomplished. In this paper, we give the results related to the first, second, and third Zagreb indices, forgotten index, hyper Zagreb index, reduced first and second Zagreb indices, multiplicative Zagreb indices, redefined version of Zagreb indices, first reformulated Zagreb index, harmonic index, atom-bond connectivity index, geometric-arithmetic index, and reduced reciprocal Randić index of a new graph operation named as “subdivision vertex-edge join” of three graphs. Full article

(This article belongs to the Special Issue Analytical and Computational Properties of Topological Indices)

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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://doi.org/10.3390/math8010098 - 7 Jan 2020

Cited by 7 | Viewed by 3123

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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22 pages, 749 KiB

Open AccessArticle

Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks

by Chang-Cheng Wei, Haidar Ali, Muhammad Ahsan Binyamin, Muhammad Nawaz Naeem and Jia-Bao Liu

Mathematics 2019, 7(4), 368; https://doi.org/10.3390/math7040368 - 23 Apr 2019

Cited by 26 | Viewed by 3814

Abstract

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical [...] Read more.

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derived a new type of graphs, which is named a third type of hex-derived networks. In our work, we discuss the third type of hex-derived networks

H D N 3 (r)

,

T H D N 3 (r)

,

R H D N 3 (r)

,

C H D N 3 (r)

, and compute exact results for topological indices which are based on degrees of end vertices. Full article

(This article belongs to the Special Issue Computational Methods in Analysis and Applications)

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16 pages, 346 KiB

Open AccessArticle

Topological Indices of mth Chain Silicate Graphs

by Jia-Bao Liu, Muhammad Kashif Shafiq, Haidar Ali, Asim Naseem, Nayab Maryam and Syed Sheraz Asghar

Mathematics 2019, 7(1), 42; https://doi.org/10.3390/math7010042 - 4 Jan 2019

Cited by 13 | Viewed by 3615

Abstract

A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are [...] Read more.

A topological index is a numerical representation of a chemical structure, while a topological descriptor correlates certain physico-chemical characteristics of underlying chemical compounds besides its numerical representation. A large number of properties like physico-chemical properties, thermodynamic properties, chemical activity, and biological activity are determined by the chemical applications of graph theory. The biological activity of chemical compounds can be constructed by the help of topological indices such as atom-bond connectivity (ABC), Randić, and geometric arithmetic (GA). In this paper, Randić, atom bond connectivity (ABC), Zagreb, geometric arithmetic (GA), ABC₄, and GA₅ indices of the mth chain silicate

S L (m, n)

network are determined. Full article

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11 pages, 261 KiB

Open AccessArticle

The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

by Fang Gao, Xiaoxin Li, Kai Zhou and Jia-Bao Liu

Mathematics 2018, 6(11), 271; https://doi.org/10.3390/math6110271 - 21 Nov 2018

Cited by 1 | Viewed by 2878

Abstract

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the [...] Read more.

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than

m

. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

18 pages, 2058 KiB

Open AccessArticle

On Degree-Based Topological Indices of Symmetric Chemical Structures

by Jia-Bao Liu, Haidar Ali, Muhammad Kashif Shafiq and Usman Munir

Symmetry 2018, 10(11), 619; https://doi.org/10.3390/sym10110619 - 9 Nov 2018

Cited by 9 | Viewed by 3808

Abstract

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph [...] Read more.

A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study HDCN1(m,n) and HDCN2(m,n) of dimension

m, n

and derive analytical closed results of general Randić index

R_{α} (G)

for different values of

α

. We also compute the general first Zagreb, ABC, GA,

A B C_{4}

and

G A_{5}

indices for these Hex derived cage networks for the first time and give closed formulas of these degree-based indices. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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20 pages, 22731 KiB

Open AccessArticle

Topological Properties of Crystallographic Structure of Molecules

by Jia-Bao Liu, Muhammad Kamran Siddiqui, Manzoor Ahmad Zahid, Muhammad Naeem and Abdul Qudair Baig

Symmetry 2018, 10(7), 265; https://doi.org/10.3390/sym10070265 - 5 Jul 2018

Cited by 22 | Viewed by 4947

Abstract

Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of [...] Read more.

Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of the crystal structure of titanium difluoride

T i F_{2}

and the crystallographic structure of cuprite

C u_{2} O

. Furthermore, we compute degree-based topological indices, mainly

A B C

,

G A

,

A B C_{4}

,

G A_{5}

and general Randić indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride

T i F_{2}

and the crystallographic structure of cuprite

C u_{2} O

. Full article

(This article belongs to the Special Issue Symmetry in Graph Theory)

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15 pages, 9185 KiB

Open AccessArticle

Topological Characterization of the Symmetrical Structure of Bismuth Tri-Iodide

by Muhammad Imran, Muhammad Arfan Ali, Sarfraz Ahmad, Muhammad Kamran Siddiqui and Abdul Qudair Baig

Symmetry 2018, 10(6), 201; https://doi.org/10.3390/sym10060201 - 4 Jun 2018

Cited by 29 | Viewed by 4473

Abstract

The bismuth tri-iodide

(B i I_{3})

is an inorganic compound. It is the result of the response of bismuth and iodine, which has inspired enthusiasm for subjective inorganic investigation. The topological indices are the numerical invariants of the molecular graph [...] Read more.

The bismuth tri-iodide

(B i I_{3})

is an inorganic compound. It is the result of the response of bismuth and iodine, which has inspired enthusiasm for subjective inorganic investigation. The topological indices are the numerical invariants of the molecular graph that portray its topology and are normally graph invariants. In 1975, Randic presented, in a bond-added substance, a topological index as a descriptor for portraying subatomic branching. In this paper, we investigate the precious stone structure of bismuth tri-iodide chain and sheet. Moreover, exact formulas of degree-based added-substance topological indices principally the first, second, and hyper Zagreb indices, the general Randic index, the geometric-arithmetic index, the fourth atom-bond connectivity index, and the fifth geometric arithmetic index of the subatomic graph of bismuth tri-iodide for both chain and sheet structures are determined. Full article

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