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Special Issue "Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory"

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

Deadline for manuscript submissions: 31 December 2023 | Viewed by 14202

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Special Issue Editors

Prof. Dr. Ismail Naci Cangul

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Guest Editor

Department of Mathematics, Faculty of Arts and Science, Bursa Uludag University, Bursa, Turkey
Interests: combinatorics; graph theory; chemical graph theory; topological indices; discrete groups; number theory

Prof. Dr. Kinkar Chandra Das

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Guest Editor

Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
Interests: spectral graph theory; extrema lGraph theory; molecular graph theory; graph labeling; graph algorithm; discrete mathematics; combinatorics
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Ahmet Sinan Cevik

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Guest Editor

Department of Mathematics, Faculty of Science, Selcuk University, Konya, Turkey
Interests: graph theory and its applications; commutative algebra; semigroup theory

Special Issue Information

Dear Colleagues,

Combinatorics is the branch of mathematics that deals with combinations of objects belonging to a finite set in accordance with certain constraints, such as those of graph theory.

A graph index is a mathematical formula that can be applied to any graph to obtain information regarding the real life problem modeled by the graph. By means of such an index, it is possible to comment on the mathematical values obtained to understand some physico-chemical properties of the molecule, or the required information on a social science problem, such as transportation or communication, which is under investigation. By employing this index, it is possible to avoid expensive and time-consuming laboratory experiments. These advantages have increased the interest in graph theory and the number of good publications in the area.

The first graph index is the Wiener index, defined and used by Harold Wiener in 1947, which helped him to compare the boiling points of some alkane isomers. Since then, more than 3000 topological graph indices have been registered in mathematical and chemical databases; that is, this subject is mostly studied by mathematicians and chemists. There are a large number of researchers who deal with topological graph indices all around the world, and the interest in the topic is rapidly increasing.

Please note that all submitted papers must be within the general scope of the Symmetry journal.

Prof. Dr. Ismail Naci Cangul
Prof. Dr. Kinkar Chandra Das
Prof. Dr. Ahmet Sinan Cevik
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

combinatorics
discrete mathematics
graph parameters
topological graph indices
regular graphs
symmetry in graphs
chemical applications of graph indices
molecular graphs
energy of graphs
graph matrices
matching
domination
independence number
derived graphs
extremal graphs
applications of graphs
chromatic graph theory
graph polynomials
labeling in graphs

Published Papers (15 papers)

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Research

Open AccessArticle

On the Hyper Zagreb Index of Trees with a Specified Degree of Vertices

Sana Shahab

and

Symmetry 2023, 15(7), 1295; https://doi.org/10.3390/sym15071295 - 21 Jun 2023

Viewed by 469

Abstract

Topological indices are the numerical descriptors that correspond to some certain physicochemical properties of a chemical compound such as the boiling point, acentric factor, enthalpy of vaporisation, heat of fusion, etc. Among these topological indices, the Hyper Zagreb index, is the most effectively used topological index to predict the acentric factor of some octane isomers. In the current work, we investigate the extremal values of the Hyper Zagreb index for some classes of trees. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Symmetry-Adapted Domination Indices: The Enhanced Domination Sigma Index and Its Applications in QSPR Studies of Octane and Its Isomers

Suha Wazzan

and

Hanan Ahmed

Symmetry 2023, 15(6), 1202; https://doi.org/10.3390/sym15061202 - 04 Jun 2023

Viewed by 593

Abstract

Molecular descriptors are essential in mathematical chemistry for studying quantitative structure–property relationships (QSPRs), and topological indices are a valuable source of information about molecular properties, such as size, cyclicity, branching degree, and symmetry. Graph theory has played a crucial role in the development of topological indices and dominating parameters for molecular descriptors. A molecule graph, under graph isomorphism conditions, represents an invariant number, and the graph theory approach considers dominating sets, which are subsets of the vertex set where every vertex outside the set is adjacent to at least one vertex inside the set. The dominating sigma index, a topological index that incorporates the mathematical principles of domination topological indices and the sigma index, is applicable to some families of graphs, such as book graphs and windmill graphs, and some graph operations, which have exact values for this new index. To evaluate the effectiveness of the domination sigma index in QSPR studies, a comparative analysis was conducted to establish an appropriate domination index that correlates with the physicochemical properties of octane and its isomers. Linear and non-linear models were developed using the QSPR approach to predict the properties of interest, and the results show that both the domination forgotten and domination first Zagreb indices exhibited satisfactory performance in comparison testing. Further research into QSAR/QSPR domination indices is required to build more robust models for predicting the physicochemical properties of organic compounds while maintaining the importance of symmetry. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Exploring the Symmetry of Curvilinear Regression Models for Enhancing the Analysis of Fibrates Drug Activity through Molecular Descriptors

Suha Wazzan

and

Nurten Urlu Ozalan

Symmetry 2023, 15(6), 1160; https://doi.org/10.3390/sym15061160 - 27 May 2023

Viewed by 539

Abstract

Quantitative structure-property relationship (QSPR) modeling is crucial in cheminformatics and computational drug discovery for predicting the activity of compounds. Topological indices are a popular molecular descriptor in QSPR modeling due to their ability to concisely capture the structural and electronic properties of molecules. Here, we investigate the use of curvilinear regression models to analyze fibrates drug activity through topological indices, which modulate lipid metabolism and improve the lipid profile. Our QSPR approach predicts the physicochemical properties of fibrates based on degrees and distances from topological indices. Our results demonstrate that topological indices can enhance the accuracy of predicting physicochemical properties and biological activities of molecules, including drugs. We also conducted density functional theory (DFT) calculations on the investigated derivatives to gain insights into their optimized geometries and electronic properties, including symmetry. The use of topological indices in QSPR modeling, which considers the symmetry of molecules, shows significant potential in improving our understanding of the structural and electronic properties of compounds. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Complexity Analysis of Benes Network and Its Derived Classes via Information Functional Based Entropies

and

Symmetry 2023, 15(3), 761; https://doi.org/10.3390/sym15030761 - 20 Mar 2023

Cited by 1 | Viewed by 1140

Abstract

The use of information–theoretical methodologies to assess graph-based systems has received a significant amount of attention. Evaluating a graph’s structural information content is a classic issue in fields such as cybernetics, pattern recognition, mathematical chemistry, and computational physics. Therefore, conventional methods for determining a graph’s structural information content rely heavily on determining a specific partitioning of the vertex set to obtain a probability distribution. A network’s entropy based on such a probability distribution is obtained from vertex partitioning. These entropies produce the numeric information about complexity and information processing which, as a consequence, increases the understanding of the network. In this paper, we study the Benes network and its novel-derived classes via different entropy measures, which are based on information functionals. We construct different partitions of vertices of the Benes network and its novel-derived classes to compute information functional dependent entropies. Further, we present the numerical applications of our findings in understanding network complexity. We also classify information functionals which describe the networks more appropriately and may be applied to other networks. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

On Neighborhood Degree-Based Topological Analysis over Melamine-Based TriCF Structure

Tony Augustine

and

Roy Santiago

Symmetry 2023, 15(3), 635; https://doi.org/10.3390/sym15030635 - 03 Mar 2023

Viewed by 908

Abstract

Triazine-based covalent organic frameworks (TriCFs) were synthesized using melamine, and cyanuric acid is a brand-new synthetic lubricant, which is thermo-stable and possesses a lamellar structure. This article demonstrates how topological descriptors for the TriCF structure are precisely evaluated using the degree sum of the end vertex neighbors and also some molecular descriptors with multiplicative neighborhood degree sums are evaluated. Furthermore, the neighborhood entropy measures for the outcomes are provided. The results are compared using the graph theoretical method. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Minimizing the Gutman Index among Unicyclic Graphs with Given Matching Number

Weijun Liu

and

Jiaqiu Wang

Symmetry 2023, 15(2), 556; https://doi.org/10.3390/sym15020556 - 20 Feb 2023

Viewed by 698

Abstract

For a connected graph G with vertex set V, denote by d(v) the degree of vertex v and d(u, v) the distance between u and v. The value [...] Read more.

For a connected graph G with vertex set V, denote by d(v) the degree of vertex v and d(u, v) the distance between u and v. The value

Gut (G) = \sum_{{u, v} \subseteq V} d (u) d (v) d (u, v)

is called the Gutman index of G. Recently, the graph minimizing the Gutman index among unicyclic graphs with pendent edges was characterized. Denoted by

U (n, m)

the set of unicyclic graphs on n vertices with matching number m. Motivated by that work, in this paper, we obtain a sharp lower bound for Gutman index of graphs in

U (n, m)

, and the extremal graph attaining the bound is also obtained. It is worth noticing that all the extremal graphs are of high symmetry, that is, they have large automorphic groups. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

On Laplacian Energy of r-Uniform Hypergraphs

N. Feyza Yalçın

Symmetry 2023, 15(2), 382; https://doi.org/10.3390/sym15020382 - 01 Feb 2023

Cited by 1 | Viewed by 818

Abstract

The matrix representations of hypergraphs have been defined via hypermatrices initially. In recent studies, the Laplacian matrix of hypergraphs, a generalization of the Laplacian matrix, has been introduced. In this article, based on this definition, we derive bounds depending pair-degree, maximum degree, and the first Zagreb index for the greatest Laplacian eigenvalue and Laplacian energy of r-uniform hypergraphs and r-uniform regular hypergraphs. As a result of these bounds, Nordhaus–Gaddum type bounds are obtained for the Laplacian energy. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

Open AccessArticle

On Trees with Given Independence Numbers with Maximum Gourava Indices

and

Symmetry 2023, 15(2), 308; https://doi.org/10.3390/sym15020308 - 22 Jan 2023

Cited by 1 | Viewed by 793

Abstract

In mathematical chemistry, molecular descriptors serve an important role, primarily in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) studies. A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about [...] Read more.

G O_{1} (N) = \sum_{u^{'} v^{'} \in E (N)} (d (u^{'}) + d (v^{'}) + d (u^{'}) d (v^{'})

) and

G O_{2} (N) = \sum_{u^{'} v^{'} \in E (N)} (d (u^{'}) + d (v^{'})) d (u^{'}) d (v^{'})

, respectively.The independence number of a graph N, being the cardinality of its maximal independent set, plays a vital role in reading the energies of chemical trees. In this research paper, it is shown that among the family of trees of order

δ

and independence number

ξ

, the spur tree denoted as

Υ_{δ, ξ}

maximizes both 1st and 2nd Gourava indices, and with these characterizations this graph is unique. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

On the Padovan Codes and the Padovan Cubes

Gwangyeon Lee

and

Jinsoo Kim

Symmetry 2023, 15(2), 266; https://doi.org/10.3390/sym15020266 - 18 Jan 2023

Cited by 1 | Viewed by 1022

Abstract

We present a new interconnection topology called the Padovan cube. Despite their asymmetric and relatively sparse interconnections, the Padovan cubes are shown to possess attractive recurrent structures. Since they can be embedded in a subgraph of the Boolean cube and can have a Fibonacci cube as a subgraph, and since they are also a supergraph of other structures, it is possible that the Padovan cubes can be useful in fault-tolerant computing. For a graph with n vertices, we characterize the Padovan cubes. We also include the number of edges, decompositions, and embeddings, as well as the diameter of the Padovan cubes. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Graphs of Wajsberg Algebras via Complement Annihilating

Necla Kırcalı Gürsoy

Symmetry 2023, 15(1), 121; https://doi.org/10.3390/sym15010121 - 01 Jan 2023

Viewed by 920

Abstract

In this paper, W-graph, called the notion of graphs on Wajsberg algebras, is introduced such that the vertices of the graph are the elements of Wajsberg algebra and the edges are the association of two vertices. In addition to this, commutative W-graphs are also symmetric graphs. Moreover, a graph of equivalence classes of Wajsberg algebra is constructed. Meanwhile, new definitions as complement annihilator and ∆-connection operator on Wajsberg algebras are presented. Lemmas and theorems on these notions are proved, and some associated results depending on the graph’s algebraic properties are presented, and supporting examples are given. Furthermore, the algorithms for determining and constructing all these new notions in each section are generated. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Gutman Connection Index of Graphs under Operations

Dalal Awadh Alrowaili

Faiz Farid

and

Muhammad Javaid

Symmetry 2023, 15(1), 21; https://doi.org/10.3390/sym15010021 - 22 Dec 2022

Viewed by 1126

Abstract

In the modern era, mathematical modeling consisting of graph theoretic parameters or invariants applied to solve the problems existing in various disciplines of physical sciences like computer sciences, physics, and chemistry. Topological indices (TIs) are one of the graph invariants which are frequently used to identify the different physicochemical and structural properties of molecular graphs. Wiener index is the first distance-based TI that is used to compute the boiling points of the paraffine. For a graph F, the recently developed Gutman Connection (GC) index is defined on all the unordered pairs of vertices as the sum of the multiplications of the connection numbers and the distance between them. In this note, the

G C

index of the operation-based symmetric networks called by first derived graph

D_{1} (F)

(subdivision graph), second derived graph

D_{2} (F)

(vertex-semitotal graph), third derived graph

D_{3} (F)

(edge-semitotal graph) and fourth derived graph

D_{4} (F)

(total graph) are computed in their general expressions consisting of various TIs of the parent graph F, where these operation-based symmetric graphs are obtained by applying the operations of subdivision, vertex semitotal, edge semitotal and the total on the graph F respectively. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

Open AccessArticle

Bounds on the General Eccentric Connectivity Index

Xinhong Yu

Muhammad Imran

Aisha Javed

Muhammad Kamran Jamil

and

Xuewu Zuo

Symmetry 2022, 14(12), 2560; https://doi.org/10.3390/sym14122560 - 04 Dec 2022

Cited by 1 | Viewed by 845

Abstract

The general eccentric connectivity index of a graph R is defined as

ξ^{e c} (R) = \sum_{u \in V (G)} d (u) e c {(u)}^{α}

, where

α

is any real number, [...] Read more.

The general eccentric connectivity index of a graph R is defined as

ξ^{e c} (R) = \sum_{u \in V (G)} d (u) e c {(u)}^{α}

, where

α

is any real number,

e c (u)

and

d (u)

represent the eccentricity and the degree of the vertex u in R, respectively. In this paper, some bounds on the general eccentric connectivity index are proposed in terms of graph-theoretic parameters, namely, order, radius, independence number, eccentricity, pendent vertices and cut edges. Moreover, extremal graphs are characterized by these bounds. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

Open AccessArticle

Characterization of Extremal Unicyclic Graphs with Fixed Leaves Using the Lanzhou Index

Dalal Awadh Alrowaili

Farwa Zafar

and

Muhammad Javaid

Symmetry 2022, 14(11), 2408; https://doi.org/10.3390/sym14112408 - 14 Nov 2022

Viewed by 990

Abstract

A topological index being a graph theoretic parameter plays a role of function for the assignment of a numerical value to a molecular graph which predicts the several physical and chemical properties of the underlying molecular graph such as heat of evaporation, critical [...] Read more.

Γ

, the Lanzhou index (Lz index) is obtained by the sum of

d e g {(v)}^{2}

d \bar{e} g (v)

over all the vertices, where

d e g (v)

and

d \bar{e} g (v)

are degrees of the vertex v in

Γ

and its complement

\bar{Γ}

respectively. Let

V_{α}^{β}

be a class of unicyclic graphs (same order and size) such that each graph of this class has order

α

and

β

leaves (vertices of degree one). In this note, we compute the lower and upper bounds of Lz index for each unicyclic graph in the class of graphs

V_{α}^{β} .

Moreover, we characterize the extremal graphs with respect to Lz index in the same class of graphs. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

On ve-Degree Irregularity Index of Graphs and Its Applications as Molecular Descriptor

Kinkar Chandra Das

and

Sourav Mondal

Symmetry 2022, 14(11), 2406; https://doi.org/10.3390/sym14112406 - 14 Nov 2022

Cited by 2 | Viewed by 1025

Abstract

Most of the molecular graphs in the area of mathematical chemistry are irregular. Therefore, irregularity measure is a crucial parameter in chemical graph theory. One such measure that has recently been proposed is the

v e

-degree irregularity index ( [...] Read more.

v e

-degree irregularity index (

i r r_{v e}

). Quantitative structure property relationship (QSPR) analysis explores the capability of an index to model numerous properties of molecules. We investigate the usefulness of the

i r r_{v e}

index in predicting different physico-chemical properties by carrying out QSPR analysis. It is established that the

i r r_{v e}

index is efficient to explain the acentric factor and boiling point of molecules with powerful accuracy. An upper bound of

i r r_{v e}

for the class of all trees is computed with identifying extremal graphs. We noticed that the result is not correct. In this report, we provide a counter example to justify our argument and determine the correct outcome. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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Open AccessArticle

Total Coloring of Some Classes of Cayley Graphs on Non-Abelian Groups

Shantharam Prajnanaswaroopa

Jayabalan Geetha

Kanagasabapathi Somasundaram

and

Teerapong Suksumran

Symmetry 2022, 14(10), 2173; https://doi.org/10.3390/sym14102173 - 17 Oct 2022

Cited by 2 | Viewed by 770

Abstract

Total Coloring of a graph G is a type of graph coloring in which any two adjacent vertices, an edge, and its incident vertices or any two adjacent edges do not receive the same color. The minimum number of colors required for the [...] Read more.

χ^{″} (G)

. Mehdi Behzad and Vadim Vizing simultaneously worked on the total colorings and proposed the Total Coloring Conjecture (TCC). The conjecture states that the maximum number of colors required in a total coloring is

Δ (G) + 2

, where

Δ (G)

is the maximum degree of the graph G. Graphs derived from the symmetric groups are robust graph structures used in interconnection networks and distributed computing. The TCC is still open for the circulant graphs. In this paper, we derive the upper bounds for

χ^{″} (G)

of some classes of Cayley graphs on non-abelian groups, typically Cayley graphs on the symmetric groups and dihedral groups. We also obtain the upper bounds of the total chromatic number of complements of Kneser graphs. Full article

(This article belongs to the Special Issue Combinatorics, Discrete Mathematics, Symmetry and Regularity in Graphs, Graph Indices, Graph Parameters and Applications of Graph Theory)

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