Index Of Graph Research Articles

Properties of the forgotten index in bipolar fuzzy graphs and applications.

Topological indices are the numbers that remain constant under graph automorphism. Topological indices describe a network's connectivity, structure, and topological characteristics. These indices have many applications in crisp graphs. However, in many cases, it is observed that some situations can't be described using the idea of crisp graphs. So, to overcome this issue, the need to define topological indices for fuzzy and bipolar fuzzy graphs arises. The F-index, or the Forgotten Index, is a significant topological index. A bipolar fuzzy graph with two opposite-sided opinions of both the edges and vertices measures the impreciseness or uncertainties of the edges and vertices along the positive and negative sides. In this article, we have presented the Forgotten Index for bipolar fuzzy graphs. Then, we have proved some theorems regarding the F-index of numerous types of bipolar fuzzy graphs, such as regular bipolar fuzzy graphs, complete bipolar fuzzy graphs, etc., the bounds of the F-index in bipolar fuzzy graphs, and the relationships of the F-index with other topological indices in bipolar fuzzy graphs. We have applied the proposed topological index, the F-index for bipolar fuzzy graphs, to matrimonial websites to find potential life partners based on compatibility and discussed the application of the Forgotten Index in gene regulatory networks.

On the mod $k$ chromatic index of graphs

For a graph $G$ and an integer $k\geq 2$, a $\chi'_{k}$-coloring of $G$ is an edge coloring of $G$ such that the subgraph induced by the edges of each color has all degrees congruent to $1 ~ (\mod k)$, and $\chi'_{k}(G)$ is the minimum number of colors in a $\chi'_{k}$-coloring of $G$. In ["The mod $k$ chromatic index of graphs is $O(k)$", J. Graph Theory. 2023; 102: 197-200], Botler, Colucci and Kohayakawa proved that $\chi'_{k}(G)\leq 198k-101$ for every graph $G$. In this paper, we show that $\chi'_{k}(G) \leq 177k-93$.

Exponential Wiener index of some silicate networks

Any graph that depicts a particular molecular structure can be given a topological graph index, also known as a molecular descriptor. This index can be used to examine numerical data and discover more about specific physical properties of molecules. Silicates are hard crystals that can be sliced into little pieces. Therefore, silicon play a vital role in many real time applications. It is the authentic material to make microchips which run in PDA, PC and other game tools. In the current study, we have determined the new topological index for a few silicate structures, including cyclic, single-chain, and Pyro silicates by grouping the vertices of the corresponding molecular graphs based on their distance sums. This index is known as the exponential Wiener index. It is expanded to include their line graphs as well. The numerical values of exponential Wiener index and multiplicative exponential Wiener index for the molecular graphs of cyclic silicates, single chain silicates and their line graphs with n vertices where n = 3, 4, 5 are explored.

Wiener and Harary index of the zero-divisor graph of ℤn

The zero-divisor graph of a ring [Formula: see text] is a graph whose vertex set consists of all the elements of [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. We study the zero-divisor graph [Formula: see text], for [Formula: see text], where [Formula: see text] and [Formula: see text] are distinct primes. Topological indices such as Wiener and Harary indices of [Formula: see text] are determined.

On Connectivity of Multidimensional Fuzzy Graphs and its Applications

Connectivity, average connectivity, and its related concepts are prominent notions in fuzzy graph theory that have applications in various areas, including neural networks, transportation problems, cluster analysis, and telecommunication. This work introduces the concepts of connectivity index, average connectivity index, type 2 connectivity index, etc. to a more generalized variant of a fuzzy graph, namely the multidimensional fuzzy graph. Basic results such as the relation of these parameters to vertices and edges, their bounds in some particular multidimensional fuzzy graphs, etc. are studied with proper illustrations. Theorems on the connectivity index of multidimensional fuzzy graphs are obtained after performing operations like direct product, tensor product, composition, join, and so on. Finally, a decision-making problem that can be effectively handled using the connectivity index is demonstrated and compared with others and similar models.

Distance and Degree Based Topological Indices of Cog-Wheel Graph, n-Barbell Graph and Boron Kagome Lattice

Abstract Let G = (V, E), be a graph with vertex and edge set, say V(G) and E(G). In chemical graphs the vertices and edges of the graphs represents atoms and chemical bond of the molecule. Since the distance based topological indices having high degree of predictability of pharmacological properties. In this paper we compute Eccentricity connectivity polynomial, Total eccentricity polynomial of Barbell graph and Cog wheel graph and degree based topological indices of Boron kagome lattice chemical structure.

Weighted Asymmetry Index: A New Graph-Theoretic Measure for Network Analysis and Optimization

Graph theory is a crucial branch of mathematics in fields like network analysis, molecular chemistry, and computer science, where it models complex relationships and structures. Many indices are used to capture the specific nuances in these structures. In this paper, we propose a new index, the weighted asymmetry index, a graph-theoretic metric quantifying the asymmetry in a network using the distances of the vertices connected by an edge. This index measures how uneven the distances from each vertex to the rest of the graph are when considering the contribution of each edge. We show how the index can capture the intrinsic asymmetries in diverse networks and is an important tool for applications in network analysis, optimization problems, social networks, chemical graph theory, and modeling complex systems. We first identify its extreme values and describe the corresponding extremal trees. We also give explicit formulas for the weighted asymmetry index for path, star, complete bipartite, complete tripartite, generalized star, and wheel graphs. At the end, we propose some open problems.

Analyzing the modified symmetric division deg index: mathematical bounds and chemical relevance

Abstract The modified symmetric division deg (MSD) index of a graph G is precisely defined as MSD ( G ) = ∑ j ∼ k 1 2 d j d k + d k d j , where d j and d k represent the degrees of j and k respectively. In this paper, we present precise bounds for the modified symmetric division deg index, expressed in the relation to the minimum and maximum degrees, the order and size of the graph, the number of pendant vertices and the minimum degree of a non-pendant vertex, the forgotten topological index, the modified second Zagreb index. We determine the upper bounds for the modified symmetric division deg index of unicyclic, bicyclic and k-cyclic graphs. Moreover, upon analyzing the modified symmetric division deg index, we observe its correlation with other well-known indices and its chemical applicability on the molecular graphs of octane isomers. At the end, we find the chemical applicability of the modified symmetric division deg index on benzenoid hydrocarbons and observe that the modified symmetric division deg index has a very strong correlation with the physical properties of benzenoid hydrocarbons.

Gutman index of fuzzy graphs with application

Gutman index of fuzzy graphs with application

A new spectral-based index for graphs and its application to polyaromatic compounds

In this study, we introduce a novel index called the modified inverse sum indeg (MISI) energy as an extension of the traditional inverse sum indeg (ISI) energy and neighbor degree sum energy. We devised an algorithm that employs the simplified molecular input line entry system (SMILES) formula of compounds to compute the MISI energy of polycyclic aromatic hydrocarbon (PAH). Additionally, we established the bounds of the MISI energy for certain classes of graphs with fixed number of vertices. A strong correlation between the MISI energy and the total [Formula: see text]-electron energy of PAHs is observed through regression analysis. Remarkably, this correlation outperforms that achieved by the classical ISI energy. These findings highlight the enhanced effectiveness of the MISI energy as a descriptor for capturing significant molecular properties. Moreover, our study establishes a foundation for further theoretical extensions and practical applications in the field of chemical graph theory.

Computational analysis of diameter eccentricity based and hyper diameter eccentricity based indices for linear saturated monocarboxylic acids

Chemical graph theory deals with chemical graphs, which are used to represent chemical systems. Topological indices of molecular graphs are an important area of research in chemical graph theory. These indices are numerical values associated with compounds that aim to establish connections between chemical structures and physical attributes, chemical reactivity, and biological activity. Many graph polynomials have been proposed to compute various topological indices. Quantitative structure-property relationship (QSPR) analysis of chemical compounds involves several regression methods that rely on these topological indices. In this article, we utilize the newly introduced DƐ-polynomial to calculate the eccentricitybased indices such as the diameter eccentricity based and hyper diameter eccentricity based index values for nineteen linear saturated monocarboxylic acids. Quantitative structureproperty relationship (QSPR) analysis is also performed for the seven thermodynamic properties of monocarboxylic acids. The relationship between thermodynamic properties and topological indices is explored using linear, quadratic and cubic regression models. The best fit curvilinear regression models were determined based on the R2 and RMSE values of the models. To further investigate the statistical significance of our models, we also conducted chi-square goodness-of-fit tests to identify the best fit models.

Some basic mathematical properties of the misbalance hadeg index of graphs

Some basic mathematical properties of the misbalance hadeg index of graphs

Some Reverse Topological Indices of Comet and Double Comet Graphs

Introduction: Chemical Graph Theory or CGT for short, is a transdisciplinary field of Mathematics wherein graphs are used to represent chemical compounds. Under this representation, the atoms of a chemical compound are expressed as vertices, while the bonds connecting these atoms are expressed as edges. Once the graph representation of a chemical compound has been determined, graph-theoretic techniques can now be used to determine various topological indices associated with the chemical compound. Topological indices are invariants of a graph that have predictive power for the chemical properties of a chemical compound. In CGT, trees, being connected and acyclic, have been an essential class of graphs. This is because, most compounds have acyclic molecular structures. In the 2023 study by Gowtham and Husin, various reverse topological indices of a family of trees called bistar graphs have been determined. Objectives: Motivated by the work of Gowtham and Husin, we determine some reverse topological indices of another family of trees called comets and double comets. Double comets are natural extension of bistar graphs. We also give a certain computational application of our results on double comet graphs. Finally, we investigate the relationship between the reverse topological indices of bistar graphs and double comets. Methods: Several important graph-theoretic concepts were used to analyze some important properties of comets and double comets, leading to the computation of their reverse vertex degree based topological indices. Results: The following reverse vertex degree-based topological indices for comets and double comets were determined: Reverse sum-connectivity, First reverse Zagreb, Second reverse Zagreb, Reverse arithmetic-geometric, Reverse geometric-arithmetic, Reverse Sombor, and Reverse Nirmala. A computational application of the results to a certain chemical compound (2,2,4,4-Tetramethylpentane) or C9H20 were also demonstrated. Conclusions: The results of the paper provided an extension to an existing result on the reverse vertex degree-based topological indices of bistar graphs by considering double comet graphs. Several topological reverse vertex degree-based topological indices were also computed for comet graphs. The computed topological indices were also applied in determining some invariants of a certain chemical compound. For future studies, we recommend that reverse vertex degree based topological indices of other family of trees will be considered.

Molecular Topology Index of a Zero Divisor Graph on a Ring of Integers Modulo Prime Power Order

In chemistry, graph theory has been widely utilized to address molecular problems, with numerous applications in graph theory and ring theory within this field. One of these applications involves topological indices that represent chemical structures with numerical values. Various types of topological indices exist, including the Wiener index, the first Zagreb index, and the hyper-Wiener index. In the context of this research, the values of the Wiener index, the first Zagreb index, and the hyper-Wiener index for zero-divisor graphs on the ring of integers modulo a prime power order will be explored through a literature review and conjecture.

Wiener index of inverse fuzzy mixed graphs with application in education system

Wiener index of inverse fuzzy mixed graphs with application in education system

Degree-based topological indices of the idempotent graph of the ring [formula omitted]

Let R be a finite commutative ring with a non-zero identity, and Id(R) be the set of idempotent elements of R. The idempotent graph of R, denoted by GId(R), is a simple undirected graph with all elements of R as vertices, and two distinct vertices u, v are adjacent if and only if u+v∈Id(R). In this paper, we consider the idempotent graph of the ring Zn and investigate some degree-based topological indices, such as the general sum-connectivity index, the general Randić index, the general Zagreb index, and the Sombor index of that graph by considering the M-polynomial of the graph.

Topology of quasi divisor graphs associated with non-associative algebra

The visualization of graphs representing algebraic structures has increasingly gained traction in chemical engineering research, emerging as a significant scientific challenge in contemporary studies. Due to their practical applications, graphs associated with groups and various algebraic systems have garnered significant interest in academic literature. Chemical graph theory employs graphs to illustrate molecular structures, where vertices symbolize atoms and edges represent bonds. The structure of quasi divisor graphs linked to non-associative algebra, including quasigroups and loops, presents numerous potential implications and wider applications. These applications span multiple disciplines, such as algebra, combinatorics, computer science, and network theory. Latin squares are tabular representations of quasigroups, generalizations of group structure. Numerous classifications of quasigroups exist in academic literature, such as Moufang quasigroups, Bol quasigroups, and alternative quasigroups. However, the structural characteristics of quasigroups exhibiting the weak inverse property closely resemble those of groups. The overall purpose of this study is to see the interrelationship between non-isomorphic quasi divisor graphs and a particular class of G-loops. To achieve our results, we employ a variety of methods, including a combinatorial approach, degree counting techniques, vertex and edge partitioning methods, as well as graph-theoretical tools and analytical procedures. In this paper, we compute a few topological indices based on degree, distance, and degree-distance using the structural aspects of quasi divisor graphs of orders 2φ,22ϕ,23ϕ and 4ϕ1ϕ2 where φ is a positive integer and ϕ1,ϕ2 are odd primes with ϕ1<ϕ2. This work also includes polynomial forms and their geometrical interpretations connected to linear, quadratic and higher degree topological indices of quasi divisor graphs associated with a class of non-commutative weak inverse property quasigroups.

A Study on Zagreb Indices of Vertex-Switching for Special Graph Classes

Many graph theorists have studied graph operations due to their applications and the advantages with heavy calculations. In a recent paper, the vertex-switching operation is analyzed, and some vertex-switched graphs are determined for some graph classes. This paper calculates the first Zagreb index and the second Zagreb index of vertex-switched star, complete bipartite, and tadpole graphs. It finally discusses the need for further research.

D-Index of Certain Ladder Graphs

For any two distinct vertices , of a connected graph , the -distance , in which the minimum is taken over all paths, and is the length of the path . The -index of is defined as . In this paper, we obtained a formula for -index or Wiener -index , where is the ladder graph, . Also, we obtained the Wiener -index and Wiener index of semi-Ladder graph .

The Multiplicative Sum Zagreb Indices of Graphs with Given Clique Number

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph G is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.