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Analytical and Computational Properties of Topological Indices

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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (31 March 2021) | Viewed by 34089

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Special Issue Editors

Prof. Dr. Eva Tourís

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Department of Mathematics, Science Faculty, Autónoma University of Madrid, Cantoblanco Campus, CP-28049 Madrid, Spain
Interests: network theory; discrete mathematics; topological indices; functions of a complex variable; potential theory; approximation theory
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Jose M. Rodriguez

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Department of Mathematics, Carlos III University of Madrid-Leganés Campus, Avenida de la Universidad 30, CP-28911, Leganés, Madrid, Spain
Interests: discrete mathematics; fractional calculus; topological indices; polynomials in graphs; geometric function theory; geometry; approximation theory
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Prof. Dr. José M. Sigarreta

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Faculty of Mathematics. Autonomous University of Guerrero-Acapulco Campus, Calle Carlos E. Adame 54, Garita, Acapulco CP-39650, Guerrero, Mexico
Interests: discrete mathematics; alliances in graphs; conformable and non-conformable calculus; geometry; topological indices
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Although Topological Indices have played an important role in Mathematical Chemistry since the seminal work of Wiener in 1947, in recent years, this role has significantly increased. On the one side, molecular descriptors constitute an aid tool in Chemistry, especially in QSPR/QSAR investigations. On the other side, they have become an important part of some areas of Mathematics, as Graph Theory; this interest has been recognized in the 2020-version of the Mathematical Subject Classification by including two new areas: 05C09 - Graphical indices (Wiener index, Zagreb index, Randić index, etc.), and 05C92 - Chemical graph theory.

The aim of this Special Issue is to attract leading researchers in this area in order to include new results on these topics, both from a theoretical and an applied point of view.

Prof. Dr. Eva Tourís
Prof. Dr. Jose M. Rodriguez
Prof. Dr. José M. Sigarreta
Guest Editors

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Keywords

Topological Indices
Graphical Indices
Chemical Graph Theory
Mathematical Chemistry
Topological Descriptors
Molecular Descriptors
Graph Optimization Problems
Polynomials on Topological Indices

Published Papers (13 papers)

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Research

11 pages, 264 KiB

Open AccessArticle

Cactus Graphs with Maximal Multiplicative Sum Zagreb Index

by Chunlei Xu, Batmend Horoldagva and Lkhagva Buyantogtokh

Symmetry 2021, 13(5), 913; https://doi.org/10.3390/sym13050913 - 20 May 2021

Cited by 4 | Viewed by 1755

Abstract

A connected graph G is said to be a cactus if any two cycles have at most one vertex in common. The multiplicative sum Zagreb index of a graph G is the product of the sum of the degrees of adjacent vertices in G. In this paper, we introduce several graph transformations that are useful tools for the study of the extremal properties of the multiplicative sum Zagreb index. Using these transformations and symmetric structural representations of some cactus graphs, we determine the graphs having maximal multiplicative sum Zagreb index for cactus graphs with the prescribed number of pendant vertices (cut edges). Furthermore, the graphs with maximal multiplicative sum Zagreb index are characterized among all cactus graphs of the given order. Full article

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

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15 pages, 331 KiB

Open AccessArticle

Bounds on the Arithmetic-Geometric Index

by José M. Rodríguez, José L. Sánchez, José M. Sigarreta and Eva Tourís

Symmetry 2021, 13(4), 689; https://doi.org/10.3390/sym13040689 - 15 Apr 2021

Cited by 12 | Viewed by 1557

Abstract

The concept of arithmetic-geometric index was recently introduced in chemical graph theory, but it has proven to be useful from both a theoretical and practical point of view. The aim of this paper is to obtain new bounds of the arithmetic-geometric index and characterize the extremal graphs with respect to them. Several bounds are based on other indices, such as the second variable Zagreb index or the general atom-bond connectivity index), and some of them involve some parameters, such as the number of edges, the maximum degree, or the minimum degree of the graph. In most bounds, the graphs for which equality is attained are regular or biregular, or star graphs. Full article

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

20 pages, 325 KiB

Open AccessArticle

Topological Indices and f-Polynomials on Some Graph Products

by Ricardo Abreu-Blaya, Sergio Bermudo, José M. Rodríguez and Eva Tourís

Symmetry 2021, 13(2), 292; https://doi.org/10.3390/sym13020292 - 09 Feb 2021

Cited by 1 | Viewed by 1830

Abstract

We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and Cartesian product, and with first Zagreb, forgotten, inverse degree and sum lordeg indices. Full article

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

12 pages, 269 KiB

Open AccessArticle

On Sombor Index

by Kinkar Chandra Das, Ahmet Sinan Çevik, Ismail Naci Cangul and Yilun Shang

Symmetry 2021, 13(1), 140; https://doi.org/10.3390/sym13010140 - 16 Jan 2021

Cited by 111 | Viewed by 6129

Abstract

The concept of Sombor index

(S O)

was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index

S O

: [...] Read more.

The concept of Sombor index

(S O)

was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index

S O

S O = S O (G) = \sum_{v_{i} v_{j} \in E (G)} \sqrt{d_{G} {(v_{i})}^{2} + d_{G} {(v_{j})}^{2}},

where

d_{G} (v_{i})

is the degree of vertex

v_{i}

in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work. Full article

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

14 pages, 291 KiB

Open AccessArticle

Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs

by Zhao Wang, Yaping Mao, Kinkar Chandra Das and Yilun Shang

Symmetry 2020, 12(10), 1711; https://doi.org/10.3390/sym12101711 - 16 Oct 2020

Cited by 8 | Viewed by 1703

Abstract

Building upon the notion of the Gutman index

SGut (G)

, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index

{SGut}_{k} (G)

of [...] Read more.

Building upon the notion of the Gutman index

SGut (G)

, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph G. The Steiner Gutman k-index

{SGut}_{k} (G)

of G is defined by

{SGut}_{k} (G)

= \sum_{S \subseteq V (G), | S | = k} (\prod_{v \in S} d e g_{G} (v)) d_{G} (S)

, in which

d_{G} (S)

is the Steiner distance of S and

d e g_{G} (v)

is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on

{SGut}_{k}

, and then investigate the Nordhaus-Gaddum-type results for the parameter

{SGut}_{k}

. We obtain sharp upper and lower bounds of

{SGut}_{k} (G) + {SGut}_{k} (\bar{G})

and

{SGut}_{k} (G) \cdot {SGut}_{k} (\bar{G})

for a connected graph G of order n, m edges, maximum degree

Δ

and minimum degree

δ

. Full article

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

14 pages, 2551 KiB

Open AccessArticle

Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index

by Wan Nor Nabila Nadia Wan Zuki, Zhibin Du, Muhammad Kamran Jamil and Roslan Hasni

Symmetry 2020, 12(10), 1591; https://doi.org/10.3390/sym12101591 - 25 Sep 2020

Cited by 4 | Viewed by 4543

Abstract

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (

A B C (G)

) and Randić index (

R (G)

) are the two most well known topological indices. Recently, Ali and Du (2017) [...] Read more.

Let G be a simple, connected and undirected graph. The atom-bond connectivity index (

A B C (G)

) and Randić index (

R (G)

) are the two most well known topological indices. Recently, Ali and Du (2017) introduced the difference between atom-bond connectivity and Randić indices, denoted as

A B C - R

index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of

A B C - R

for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for

A B C - R

index of such trees with given number of pendant vertices. The role of symmetry has great importance in different areas of graph theory especially in chemical graph theory. Full article

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

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16 pages, 732 KiB

Open AccessFeature PaperArticle

Computational Properties of General Indices on Random Networks

by R. Aguilar-Sánchez, I. F. Herrera-González, J. A. Méndez-Bermúdez and José M. Sigarreta

Symmetry 2020, 12(8), 1341; https://doi.org/10.3390/sym12081341 - 11 Aug 2020

Cited by 16 | Viewed by 1857

Abstract

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices,

M_{1}^{α} (G)

and

M_{2}^{α} (G)

, and the general sum-connectivity index,

χ_{α} (G)

) [...] Read more.

We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices,

M_{1}^{α} (G)

and

M_{2}^{α} (G)

, and the general sum-connectivity index,

χ_{α} (G)

) as well as of general versions of indices of interest: the general inverse sum indeg index

I S I_{α} (G)

and the general first geometric-arithmetic index

G A_{α} (G)

(with

α \in R

). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks

G_{ER} (n_{ER}, p)

and random geometric (RG) graphs

G_{RG} (n_{RG}, r)

. The ER random networks are formed by

n_{ER}

vertices connected independently with probability

p \in [0, 1]

; while the RG graphs consist of

n_{RG}

vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius

r \in [0, \sqrt{2}]

. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree

⟨k⟩

of the corresponding random network models, where

⟨k_{ER}⟩ = (n_{ER} - 1) p

and

⟨k_{RG}⟩ = (n_{RG} - 1) (π r^{2} - 8 r^{3} / 3 + r^{4} / 2)

. That is,

⟨X (G_{ER})⟩ / n_{ER} \approx ⟨X (G_{RG})⟩ / n_{RG}

⟨k_{ER}⟩ = ⟨k_{RG}⟩

, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks. Full article

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

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23 pages, 640 KiB

Open AccessArticle

Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks

by Usman Ali, Muhammad Javaid and Abdulaziz Mohammed Alanazi

Symmetry 2020, 12(8), 1320; https://doi.org/10.3390/sym12081320 - 07 Aug 2020

Cited by 11 | Viewed by 2041

Abstract

Gutman and Trinajstić (1972) defined the connection-number based Zagreb indices, where connection number is degree of a vertex at distance two, in order to find the electron energy of alternant hydrocarbons. These indices remain symmetric for the isomorphic (molecular) networks. For the prediction of physicochemical and symmetrical properties of octane isomers, these indices are restudied in 2018. In this paper, first and second Zagreb connection coindices are defined and obtained in the form of upper bounds for the resultant networks in the terms of different indices of their factor networks, where resultant networks are obtained from two networks by the product-related operations, such as cartesian, corona, and lexicographic. For the molecular networks linear polynomial chain, carbon nanotube, alkane, cycloalkane, fence, and closed fence, first and second Zagreb connection coindices are computed in the consequence of the obtained results. An analysis of Zagreb connection indices and coindices on the aforesaid molecular networks is also included with the help of their numerical values and graphical presentations that shows the symmetric behaviour of these indices and coindices with in certain intervals of order and size of the under study (molecular) networks. Full article

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

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12 pages, 268 KiB

Open AccessArticle

Further Theory of Neutrosophic Triplet Topology and Applications

by Mohammed A. Al Shumrani, Muhammad Gulistan and Florentin Smarandache

Symmetry 2020, 12(8), 1207; https://doi.org/10.3390/sym12081207 - 23 Jul 2020

Cited by 4 | Viewed by 1563

Abstract

In this paper we study and develop the Neutrosophic Triplet Topology (NTT) that was recently introduced by Sahin et al. Like classical topology, the NTT tells how the elements of a set relate spatially to each other in a more comprehensive way using the idea of Neutrosophic Triplet Sets. This article is important because it opens new ways of research resulting in many applications in different disciplines, such as Biology, Computer Science, Physics, Robotics, Games and Puzzles and Fiber Art etc. Herein we study the application of NTT in Biology. The Neutrosophic Triplet Set (NTS) has a natural symmetric form, since this is a set of symmetric triplets of the form <A>, <anti(A)>, where <A> and <anti(A)> are opposites of each other, while <neuti(A)>, being in the middle, is their axis of symmetry. Further on, we obtain in this paper several properties of NTT, like bases, closure and subspace. As an application, we give a multicriteria decision making for the combining effects of certain enzymes on chosen DNA using the developed theory of NTT. Full article

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

15 pages, 266 KiB

Open AccessArticle

New Bounds for Topological Indices on Trees through Generalized Methods

by Álvaro Martínez-Pérez and José M. Rodríguez

Symmetry 2020, 12(7), 1097; https://doi.org/10.3390/sym12071097 - 02 Jul 2020

Cited by 2 | Viewed by 1608

Abstract

Topological indices are useful for predicting the physicochemical behavior of chemical compounds. A main problem in this topic is finding good bounds for the indices, usually when some parameters of the graph are known. The aim of this paper is to use a unified approach in order to obtain several new inequalities for a wide family of topological indices restricted to trees and to characterize the corresponding extremal trees. The main results give upper and lower bounds for a large class of topological indices on trees, fixing or not the maximum degree. This class includes the first variable Zagreb, the Narumi–Katayama, the modified Narumi–Katayama and the Wiener index. Full article

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

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 15 | Viewed by 1982

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 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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16 pages, 3207 KiB

Open AccessArticle

M-Polynomial and Degree Based Topological Indices of Some Nanostructures

by Zahid Raza and Mark Essa K. Sukaiti

Symmetry 2020, 12(5), 831; https://doi.org/10.3390/sym12050831 - 19 May 2020

Cited by 46 | Viewed by 4205

Abstract

The association of M-polynomial to chemical compounds and chemical networks is a relatively new idea, and it gives good results about the topological indices. These results are then used to correlate the chemical compounds and chemical networks with their chemical properties and bioactivities. In this paper, an effort is made to compute the general form of the M-polynomials for two classes of dendrimer nanostars and four types of nanotubes. These nanotubes have very nice symmetries in their structural representations, which have been used to determine the corresponding M-polynomials. Furthermore, by using the general form of M-polynomial of these nanostructures, some degree-based topological indices have been computed. In the end, the graphical representation of the M-polynomials is shown, and a detailed comparison between the obtained topological indices for aforementioned chemical structures is discussed. Full article

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

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9 pages, 759 KiB

Open AccessArticle

Remarks on Distance Based Topological Indices for ℓ-Apex Trees

by Martin Knor, Muhammad Imran, Muhammad Kamran Jamil and Riste Škrekovski

Symmetry 2020, 12(5), 802; https://doi.org/10.3390/sym12050802 - 12 May 2020

Cited by 2 | Viewed by 1878

Abstract

A graph G is called an ℓ-apex tree if there exist a vertex subset

A \subset V (G)

with cardinality ℓ such that

G - A

is a tree and there is no other subset of smaller cardinality with this property. [...] Read more.

A graph G is called an ℓ-apex tree if there exist a vertex subset

A \subset V (G)

with cardinality ℓ such that

G - A

is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for

ℓ = 1

and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs. Full article

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

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