Special Issue "Graph Theory at Work in Carbon Chemistry"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (30 April 2020).

Special Issue Editors

Prof. Ottorino Ori
E-Mail Website
Guest Editor
Actinium Chemical Research, Rome, Italy
Interests: topology; fullerenes; complex systems; graphene; carbon nanomaterials
Prof. Dr. Boris Furtula
E-Mail Website
Guest Editor
Faculty of Science, University of Kragujevac, P. O. Box 60 34000 Kragujevac, Serbia
Interests: chemical graph theory; investigation of molecular descriptors' properties; theoretical study of electronic structure of polycyclic aromatic compounds

Special Issue Information

Dear Colleagues,

Carbon allotropes are basically distinguished by the way in which carbon atoms are linked to each other, forming different types of networks (graphs) of carbon atoms. Different structures are builds with sp2-hybridized carbon atoms like PAHs, graphite, nanotubes, nanocones, nanohorns, and fullerenes. In these chemical systems, each node has degree 3, the basic hexagonal rings are connected with other similar rings in various topological ways, and in some cases, with the introduction of some pentagonal rings. By topology, the number of 5-rings is limited to 12. The resulting nanostructures exhibit different dimensionality (ranging from 0 to 2, including fractals dimensions) and genus, being planar, spherical, toroidal surfaces equally represented among the sp2-carbon systems.

In sp3-diamond, every atom in the carbon now has a valency of four, and the carbon networks increase their dimensionality to 3. The influence of topology on the properties of molecular and crystal structures can be even more appreciated when defects are introduced in the systems. Topological indices are frequently employed as tools in various chemical investigations. They are carrying information on the chemical structures, rationalizing in this way the relationships between the structural formula of a molecule and its physicochemical and/or biological properties. Thence, the knowledge of topological molecular descriptors (which structural features they are capturing, the relations among them, their bounds, etc.) is crucially important for their proper usage in chemical applications.

This issue is devoted to the contemporary applications of chemical graph theory tools in modeling the carbon-based molecular structures and the investigations of topological molecular descriptors and their qualities.

Prof. Ottorino Ori
Prof. Dr. Boris Furtula
Guest Editor

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Keywords

  • Topological properties of low dimensional systems
  • Topological symmetry
  • Generalized Stone–Wales transformations
  • Defective graphs
  • Topological descriptors
  • Bounds of topological descriptors

Published Papers (10 papers)

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Research

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Article
Wiener–Hosoya Matrix of Connected Graphs
Mathematics 2021, 9(4), 359; https://doi.org/10.3390/math9040359 - 11 Feb 2021
Viewed by 367
Abstract
Let G be a connected (molecular) graph with the vertex set V(G)={v1,,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1in. 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×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj 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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Article
Skeletal Rearrangements of the C240 Fullerene: Efficient Topological Descriptors for Monitoring Stone–Wales Transformations
Mathematics 2020, 8(6), 968; https://doi.org/10.3390/math8060968 - 12 Jun 2020
Cited by 3 | Viewed by 672
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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Article
On the Aα-Spectral Radii of Cactus Graphs
Mathematics 2020, 8(6), 869; https://doi.org/10.3390/math8060869 - 28 May 2020
Viewed by 556
Abstract
Let A ( G ) be the adjacent matrix and D ( G ) the diagonal matrix of the degrees of a graph G, respectively. For 0 α 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 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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Article
Wiener Index of Edge Thorny Graphs of Catacondensed Benzenoids
Mathematics 2020, 8(4), 467; https://doi.org/10.3390/math8040467 - 27 Mar 2020
Cited by 1 | Viewed by 2520
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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Article
Novel Face Index for Benzenoid Hydrocarbons
Mathematics 2020, 8(3), 312; https://doi.org/10.3390/math8030312 - 01 Mar 2020
Cited by 3 | Viewed by 981
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 F I ( G ) = f F ( G ) d ( f ) = v f , f 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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Article
Some Bounds on Zeroth-Order General Randić Index
Mathematics 2020, 8(1), 98; https://doi.org/10.3390/math8010098 - 07 Jan 2020
Cited by 2 | Viewed by 591
Abstract
For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = u 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 ) = u V ( G ) d ( u ) γ , where γ 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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Article
Enumeration of Pentahexagonal Annuli in the Plane
Mathematics 2019, 7(12), 1156; https://doi.org/10.3390/math7121156 - 01 Dec 2019
Viewed by 690
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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Article
Total and Double Total Domination Number on Hexagonal Grid
Mathematics 2019, 7(11), 1110; https://doi.org/10.3390/math7111110 - 15 Nov 2019
Cited by 1 | Viewed by 749
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 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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Article
On the Wiener Complexity and the Wiener Index of Fullerene Graphs
Mathematics 2019, 7(11), 1071; https://doi.org/10.3390/math7111071 - 07 Nov 2019
Cited by 5 | Viewed by 946
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 232 and IPR fullerene graphs of order n 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

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Review
Properties of Entropy-Based Topological Measures of Fullerenes
Mathematics 2020, 8(5), 740; https://doi.org/10.3390/math8050740 - 07 May 2020
Cited by 5 | Viewed by 741
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 Ia(G), the symmetry index, a degree-based entropy measure Iλ(G), the eccentric-entropy Ifσ(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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