Computing Exact Values for Gutman Indices of Sum Graphs under Cartesian Product

A fuzzy graph is one of the versatile application tools in the field of mathematics, which allows the user to easily describe the fuzzy relation between any objects. The nature of fuzziness is favorable for any environment, which supports to predict the problem and solving it. Fuzzy graphs are beneficial to give more precision and flexibility to the system as compared to the classical model (i.e.,) crisp theory. A topological index is a numerical quantity for the structural graph of the molecule and it can be represented through Graph theory. Moreover, its application not only in the field of chemistry can also be applied in areas including computer science, networking, etc. A lot of topological indices are available in chemical-graph theory and H. Wiener proposed the first index to estimate the boiling point of alkanes called ‘Wiener index’. Many topological indices exist only in the crisp but it’s new to the fuzzy graph environment. The main aim of this paper is to define the topological indices in fuzzy graphs. Here, indices defined in fuzzy graphs are Modified Wiener index, Hyper Wiener index, Schultz index, Gutman index, Zagreb indices, Harmonic index, and Randić index with illustrations. Bounds for some of the indices are proved. The algorithms for distance matrix and MWI are shown. Finally, the application of these indices is discussed.

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.

We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we present three families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes.Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal.

Fiber-complemented graphs — I: structure and invariant subgraphs

A graph G is k-factor-critical if G ? S has a perfect matching for any k-subset S of V(G). In this paper, we investigate the factor-criticality in Cartesian products of graphs and show that Cartesian product of an m-factor-critical graph and an n-factor-critical graph is $${(m+n+\varepsilon )}$$ -factor-critical, where $${\varepsilon = 0}$$ if both of m and n are even; $${\varepsilon = 1}$$ , otherwise. Moreover, this result is best possible.

PM_{2.5}, a major air pollutant, often carries toxic components such as polycyclic aromatic hydrocarbons (PAHs), which pose potential health risks. This study investigated the relationships between five physicochemical properties-boiling point, molecular mass, enthalpy of vaporization, molar volume, and surface tension-of 15 PAHs and seven molecular topological indices (including the Wiener, Hyper-Wiener, Distance-degree, and Gutman indices) using regression analysis. Results indicated that the selected topological indices were strongly correlated with the properties of PAHs, with correlation coefficients generally exceeding 0.7. In particular, quadratic regression models based on the Distance-degree and Gutman indices showed a very high correlation with enthalpy of vaporization ($R = 0.998$). This work provides quantitative insights into the structure–property relationships of PAHs and may support further theoretical developments in the risk assessment and source control of PM_{2.5}-bound PAHs.

On normality of n-Cayley graphs

Pseudo-one-factorizations of regular graphs of odd order — II. products of graphs

Chemical graph theory is a sub-branch of mathematical chemistry, assuming each atom of a molecule is a vertex and each bond between atoms as an edge. Owing to this theory, it is possible to avoid the difficulties of chemical analysis because many of the chemical properties of molecules can be determined and analyzed via topological indices. Due to these parameters, it is possible to determine the physicochemical properties, biological activities, environmental behaviours and spectral properties of molecules. Nowadays, studies on the zero divisor graph of Z_n via topological indices is a trending field in spectral graph theory. For a commutative ring R with identity, the prime ideal sum graph of R is a graph whose vertices are nonzero proper ideals of R and two distinctvertices I and J are adjacent if and only if I+J is a prime ideal of R. In this study the forgotten topological index and Wiener index of the prime ideal sum graph of Z_n are calculated for n=p^α,pq,p^2 q,p^2 q^2,pqr,p^3 q,p^2 qr,pqrs where p,q,r and s are distinct primes and a Sage math code is developed for designing graph and computing the indices. In the light of this study, it is possible to handle the other topological descriptors for computing and developing new algorithms for next studies and to study some spectrum and graph energies of certain finite rings with respect to PIS-graph easily.

Estimation of Laplacian spectra of direct and strong product graphs

This work presents the employment of graph theory for cryptographic applications. A Cartesian product graph (CPG) is used to propose new symmetric encryption schemes. On proposed encryption schemes, the ciphertexts of the plaintexts are sent to the receiver entity as the CPGs. The CPG is created based on two graphs G and H that are generated depending on the random choices of the plaintext bits. These bits are used as vertices of graphs G and H. The purpose of using the CPG is to increase the security level of the symmetric encryption schemes in comparison with other encryption schemes. Two study cases of the proposed symmetric Cartesian product graphic encryption (SCPGE) schemes are presented as new experimental results. The security issues of the proposed SCPGE schemes are determined. The SCPGE schemes consider as new insights for more secure communications.

We give two combinatorial proofs of a product formula for the number of spanning trees of the $n$-dimensional hypercube. The first proof is based on the assertion that if one chooses a uniformly random rooted spanning tree of the hypercube and orient each edge from parent to child, then the parallel edges of the hypercube get orientations which are independent of one another. This independence property actually holds in a more general context and has intriguing consequences. The second proof uses some "killing involutions'' in order to identify the factors in the product formula. It leads to an enumerative formula for the spanning trees of the $n$-dimensional hypercube augmented with diagonals edges, counted according to the number of edges of each type. We also discuss more general formulas, obtained using a matrix-tree approach, for the number of spanning trees of the Cartesian product of complete graphs.

Chemical graph theory has an important role in the development of chemical sciences. A graph is produced from certain molecular structure by means of applying several graphical operations. The local graph parameter is valency, which is defined for every vertex as the number associates with other vertices in a graph, for example an atom in a molecule. The demonstration of chemical networks and chemical compounds with the help of M-polynomials is a novel idea. The M-polynomial of different molecular structures help to compute several topological indices. A topological index is a numeric quantity that describes the whole structure of a molecular graph of the chemical compound and clarifies its physical features, chemical reactivates and boiling activities. In this paper we computed M-Polynomial and topological indices of TUHRC4 Graph, then we recovered numerous topological indices using the M-polynomials.

The Y-index of a graph is defined by the sum of four of degrees of the vertices of a graph. Among the all topological indices the Zagreb indices have been used more considerably than any other topological indices in chemical literature. The concept of Corona Product is a recent inclusion to mathematical vocabulary. One of the most significant graph operations is the corona product of several generic and specific graphs, which is one of the most well-known graph products. In this study, we derive some explicit formulations of several corona product types, including subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighborhood corona, subdivision-edge neighborhood corona, and vertex-edge corona of two graphs.

Topological molecular descriptor is calculated by a mathematical procedure from the structure of chemical compounds represented by a molecular graph, containing information about their structural characteristics [1]. There are many defined topological descriptors in literature which transform specific chemical information into useful numerical values that have been used for correlation of structure-descriptors with various physico-chemical properties [2]. Among the first topological descriptors that accelerate further development of the field are M1 and M2 indices introduced by Gutman and Trinajstić in 1972 [3] (later named Zagreb indices). Vertices of a molecular graph represent atoms and edges connecting vertices of a graph represent carbon-carbon (or non-hydrogen) chemical bonds. The total number of edges connecting one vertex with its (first) neighbour carbon vertices corresponds to valence of that atom with other carbons within a molecule. One of the largest classes of topological descriptors is the one based on the analysis of vertex degrees of a molecular graph [4]. Starting from electronic version of chemical structures we will develop and optimize an application for calculation of different vertex-degree-based topological descriptors, which is made available for free use to the research community at http://meteo2.irb.hr/indexer.