Special Issue "Analytical and Computational Properties of Topological Indices"

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

Deadline for manuscript submissions: 31 March 2021.

Special Issue Editors

Prof. Dr. Eva Tourís
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Guest Editor
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
Prof. Dr. Jose M. Rodriguez
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Guest Editor
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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Guest Editor
Faculty of Mathematics. Autonomous University of Guerrero-Acapulco Campus, Calle Carlos E. Adame 54, Garita, CP-39650, Acapulco, Guerrero, Mexico
Interests: discrete mathematics; alliances in graphs; conformable and non-conformable calculus; geometry; topological indices
Special Issues and Collections 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

Manuscript Submission Information

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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 (9 papers)

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Research

Open AccessArticle
Nordhaus–Gaddum-Type Results for the Steiner Gutman Index of Graphs
Symmetry 2020, 12(10), 1711; https://doi.org/10.3390/sym12101711 - 16 Oct 2020
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 SGutk(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 SGutk(G) of G is defined by SGutk(G)=SV(G),|S|=kvSdegG(v)dG(S), in which dG(S) is the Steiner distance of S and degG(v) is the degree of v in G. In this paper, we derive new sharp upper and lower bounds on SGutk, and then investigate the Nordhaus-Gaddum-type results for the parameter SGutk. We obtain sharp upper and lower bounds of SGutk(G)+SGutk(G¯) and SGutk(G)·SGutk(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)
Open AccessArticle
Extremal Trees with Respect to the Difference between Atom-Bond Connectivity Index and Randić Index
Symmetry 2020, 12(10), 1591; https://doi.org/10.3390/sym12101591 - 25 Sep 2020
Abstract
Let G be a simple, connected and undirected graph. The atom-bond connectivity index (ABC(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 (ABC(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 ABCR index. In this paper, we determine the fourth, the fifth and the sixth maximum chemical trees values of ABCR for chemical trees, and characterize the corresponding extremal graphs. We also obtain an upper bound for ABCR 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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Open AccessFeature PaperArticle
Computational Properties of General Indices on Random Networks
Symmetry 2020, 12(8), 1341; https://doi.org/10.3390/sym12081341 - 11 Aug 2020
Cited by 1
Abstract
We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1α(G) and M2α(G), and the general sum-connectivity index, χα(G)) [...] Read more.
We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1α(G) and M2α(G), and the general sum-connectivity index, χα(G)) as well as of general versions of indices of interest: the general inverse sum indeg index ISIα(G) and the general first geometric-arithmetic index GAα(G) (with αR). We apply these indices on two models of random networks: Erdös–Rényi (ER) random networks GER(nER,p) and random geometric (RG) graphs GRG(nRG,r). The ER random networks are formed by nER vertices connected independently with probability p[0,1]; while the RG graphs consist of nRG 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[0,2]. 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 kER=(nER1)p and kRG=(nRG1)(πr28r3/3+r4/2). That is, X(GER)/nERX(GRG)/nRG if kER=kRG, 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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Open AccessArticle
Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks
Symmetry 2020, 12(8), 1320; https://doi.org/10.3390/sym12081320 - 07 Aug 2020
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 [...] Read more.
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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Open AccessArticle
Further Theory of Neutrosophic Triplet Topology and Applications
Symmetry 2020, 12(8), 1207; https://doi.org/10.3390/sym12081207 - 23 Jul 2020
Cited by 1
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 [...] Read more.
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)
Open AccessArticle
New Bounds for Topological Indices on Trees through Generalized Methods
Symmetry 2020, 12(7), 1097; https://doi.org/10.3390/sym12071097 - 02 Jul 2020
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 [...] Read more.
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)
Open AccessArticle
On Valency-Based Molecular Topological Descriptors of Subdivision Vertex-Edge Join of Three Graphs
Symmetry 2020, 12(6), 1026; https://doi.org/10.3390/sym12061026 - 17 Jun 2020
Cited by 3
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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Open AccessArticle
M-Polynomial and Degree Based Topological Indices of Some Nanostructures
Symmetry 2020, 12(5), 831; https://doi.org/10.3390/sym12050831 - 19 May 2020
Cited by 5
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. [...] Read more.
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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Open AccessArticle
Remarks on Distance Based Topological Indices for -Apex Trees
Symmetry 2020, 12(5), 802; https://doi.org/10.3390/sym12050802 - 12 May 2020
Abstract
A graph G is called an ℓ-apex tree if there exist a vertex subset A V ( G ) with cardinality 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 V ( G ) with cardinality 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 = 1 and pose some open questions for higher . 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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