Research

16 pages, 671 KiB

Open AccessArticle

On the Entire Harmonic Index and Entire Harmonic Polynomial of Graphs

by Anwar Saleh and Samirah H. Alsulami

Symmetry 2024, 16(2), 208; https://doi.org/10.3390/sym16020208 - 09 Feb 2024

Viewed by 629

A topological descriptor is a numerical parameter that describes a chemical structure using the related molecular graph. Topological descriptors have significance in mathematical chemistry, particularly for studying QSPR and QSAR. In addition, if a topological descriptor has a reciprocal link with a molecular [...] Read more.

A topological descriptor is a numerical parameter that describes a chemical structure using the related molecular graph. Topological descriptors have significance in mathematical chemistry, particularly for studying QSPR and QSAR. In addition, if a topological descriptor has a reciprocal link with a molecular attribute, it is referred to as a topological index. The use of topological indices can help to examine the physicochemical features of chemical compounds because they encode certain attributes of a molecule. The Randić index is a molecular structure descriptor that has several applications in chemistry and medicine. In this paper, we introduce a new version of the Randić index to the inclusion of the intermolecular forces between bonds with atoms, referred to as an entire Harmonic index (EHI), and we present the entire Harmonic polynomial (EHP) of a graph. Specific formulas have been obtained for certain graph classes, and graph operations have been obtained. Bounds and some important results have been found. Furthermore, we demonstrate that the correlation coefficients for the new index lie between

0.909

and 1. In the context of enthalpy of formation and

π

-electronic energy, the acquired values are significantly higher than those observed for the Harmonic index and the Randić index. 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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21 pages, 1949 KiB

Open AccessArticle

Computing Topological Descriptors of Prime Ideal Sum Graphs of Commutative Rings

by Esra Öztürk Sözen, Turki Alsuraiheed, Cihat Abdioğlu and Shakir Ali

Symmetry 2023, 15(12), 2133; https://doi.org/10.3390/sym15122133 - 30 Nov 2023

Viewed by 1878

Abstract

Let

n \geq 1

be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of

Z_{n}

, where [...] Read more.

Let

n \geq 1

be a fixed integer. The main objective of this paper is to compute some topological indices and coindices that are related to the graph complement of the prime ideal sum (PIS) graph of

Z_{n}

, where

n = p^{α}, p^{2} q, p^{2} q^{2}, p q r, p^{3} q,

p^{2} q r

, and

p q r s

for the different prime integers

p, q, r,

and s. Moreover, we construct M-polynomials and

C o M

-polynomials using the

P I S

-graph structure of

Z_{n}

to avoid the difficulty of computing the descriptors via formulas directly. Furthermore, we present a geometric comparison for representations of each surface obtained by M-polynomials and

C o M

-polynomials. Finally, we discuss the applicability of algebraic graphs to chemical graph theory. 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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15 pages, 1518 KiB

Open AccessArticle

Characterizing Interconnection Networks in Terms of Complexity via Entropy Measures

by Jinhong Zhang, Asfand Fahad, Muzammil Mukhtar and Ali Raza

Symmetry 2023, 15(10), 1868; https://doi.org/10.3390/sym15101868 - 04 Oct 2023

Cited by 2 | Viewed by 727

Abstract

One of the most recent advancements in graph theory is the use of a multidisciplinary approach to the investigation of specific structural dependent features, such as physico-chemical properties, biological activity and the entropy measure of a graph representing objects like a network or [...] Read more.

One of the most recent advancements in graph theory is the use of a multidisciplinary approach to the investigation of specific structural dependent features, such as physico-chemical properties, biological activity and the entropy measure of a graph representing objects like a network or a chemical compound. The ability of entropy measures to determine both the certainty and uncertainty about objects makes them one of the most investigated topics in science along with its multidisciplinary nature. As a result, many formulae, based on vertices, edges and symmetry, for determining the entropy of graphs have been developed and investigated in the field of graph theory. These measures assist in understanding the characteristics of graphs, such as the complexity of the networks or graphs, which may be determined using entropy measures. In this paper, we derive formulae of entropy measures of an extensively studied family of the interconnection networks and classify them in terms of complexity. This is accomplished by utilizing all three tools, including analytical formulae, graphical methods and numerical tables. 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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11 pages, 322 KiB

Open AccessArticle

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

by Muhammad Rizwan, Sana Shahab, Akhlaq Ahmad Bhatti, Muhammad Javaid and Mohd Anjum

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

Viewed by 878

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 [...] Read more.

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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32 pages, 973 KiB

Open AccessArticle

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

by Suha Wazzan and Hanan Ahmed

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

Cited by 2 | Viewed by 1012

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 [...] Read more.

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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22 pages, 1825 KiB

Open AccessArticle

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

by Suha Wazzan and Nurten Urlu Ozalan

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

Cited by 1 | Viewed by 925

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. [...] Read more.

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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18 pages, 716 KiB

Open AccessArticle

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

by Jun Yang, Asfand Fahad, Muzammil Mukhtar, Muhammad Anees, Amir Shahzad and Zahid Iqbal

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

Cited by 5 | Viewed by 1670

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 [...] Read more.

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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16 pages, 1562 KiB

Open AccessArticle

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

by Tony Augustine and Roy Santiago

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

Cited by 3 | Viewed by 1269

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 [...] Read more.

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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10 pages, 259 KiB

Open AccessArticle

Minimizing the Gutman Index among Unicyclic Graphs with Given Matching Number

by Weijun Liu and Jiaqiu Wang

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

Cited by 1 | Viewed by 1004

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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12 pages, 294 KiB

Open AccessArticle

On Laplacian Energy of r-Uniform Hypergraphs

by N. Feyza Yalçın

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

Cited by 2 | Viewed by 1246

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 [...] Read more.

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)

9 pages, 2459 KiB

Open AccessArticle

On Trees with Given Independence Numbers with Maximum Gourava Indices

by Ying Wang, Adnan Aslam, Nazeran Idrees, Salma Kanwal, Nabeela Iram and Asima Razzaque

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

Cited by 2 | Viewed by 1131

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.

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 its size, symmetry, degree of branching, and cyclicity. For any graph N, the first and second Gourava indices are defined as

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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14 pages, 548 KiB

Open AccessArticle

On the Padovan Codes and the Padovan Cubes

by Gwangyeon Lee and Jinsoo Kim

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

Cited by 1 | Viewed by 1375

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 [...] Read more.

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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19 pages, 351 KiB

Open AccessArticle

Graphs of Wajsberg Algebras via Complement Annihilating

by Necla Kırcalı Gürsoy

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

Viewed by 1171

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 [...] Read more.

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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12 pages, 251 KiB

Open AccessArticle

Gutman Connection Index of Graphs under Operations

by Dalal Awadh Alrowaili, Faiz Farid and Muhammad Javaid

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

Cited by 1 | Viewed by 1435

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 [...] Read more.

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)

9 pages, 263 KiB

Open AccessArticle

Bounds on the General Eccentric Connectivity Index

by 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 1160

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)

7 pages, 265 KiB

Open AccessArticle

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

by Dalal Awadh Alrowaili, Farwa Zafar and Muhammad Javaid

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

Cited by 1 | Viewed by 1372

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.

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 temperature, surface tension, boiling point, octanol-water partition coefficient, density and flash points. For a (molecular) graph

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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12 pages, 1948 KiB

Open AccessArticle

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

by Kinkar Chandra Das and Sourav Mondal

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

Cited by 3 | Viewed by 1371

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.

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 (

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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9 pages, 293 KiB

Open AccessArticle

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

by 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 1156

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.

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 total coloring of a graph is called the total chromatic number of the graph, denoted by

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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