The field of mathematics that studies the relationship between algebraic structures and graphs is known as algebraic graph theory. It incorporates concepts from graph theory, which examines the characteristics and topology of graphs, with those from abstract algebra, which deals with algebraic structures such as groups, rings, and fields. If the vertex set of a graph $\widehat{G}$ is fully made up of the zero divisors of the modular ring ${\mathbb{Z}}_n$, the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (modn), they are regarded as neighbors. Entropy, a notion taken from information theory and used in graph theory, measures the degree of uncertainty or unpredictability associated with a graph or its constituent elements. Entropy measurements may be used to calculate the structural complexity and information complexity of graphs. The first, second and second modified Zagrebs, general and inverse general Randics, third and fifth symmetric divisions, harmonic and inverse sum indices, and forgotten topological indices are a few topological indices that are examined in this article for particular families of zero-divisor graphs. A numerical and graphical comparison of computed topological indices over a proposed structure has been studied. Furthermore, different kinds of entropies, such as the first, second, and third redefined Zagreb, are also investigated for a number of families of zero-divisor graphs.

Fuzzy graph models are present everywhere from natural to artificial structures, embodying the dynamic processes in physical, biological, and social systems. As real-life problems are often uncertain on account of inconsistent and indeterminate information, it seems very demanding for an expert to model those problems using a fuzzy graph. To deal with the uncertainty associated with the inconsistent and indeterminate information of any real-world problems, a neutrosophic graph can be applied, where fuzzy graphs may not bear any fruitful results. The past definitions limitations in fuzzy graphs have directed us to present new definitions in single-valued neutrosophic graph (SVNG). A SVNG has several applications in the fields of physics, bio and connectivity of socialism. It has been an advantageous scope in the recent times for providing such information which is incomplete or uncertain accounting in real problems that gives the direction to describe the relationship between nodes. Operations are conveniently used in many combinatorial applications. In various situations, they present a suitable construction means; therefore, the current study, seeks to present and explore the key features of new operations, including: rejection, maximal product, symmetric difference, and residue product of SVNG. We have discuss the concept of maximal product on two strong-(SVNGS) and maximal product of connected-SVNG with examples. This research article presents the notions of degree of a vertex and total degree of a vertex in SVNG. Moreover, this study summarizes the specific conditions needed for obtaining vertices degrees in SVNG under the operations of maximal product, symmetric difference, residue product, and rejection. In addition, an application was illustrated in the food and agriculture organization with an algorithm to emphasize the contributions of this research article in practical applications.

Entropy is a thermodynamic function in physics that measures the randomness and disorder of molecules in a particular system or process based on the diversity of configurations that molecules might take. Distance-based entropy is used to address a wide range of problems in the domains of mathematics, biology, chemical graph theory, organic and inorganic chemistry, and other disciplines. We explain the basic applications of distance-based entropy to chemical phenomena. These applications include signal processing, structural studies on crystals, molecular ensembles, and quantifying the chemical and electrical structures of molecules. In this study, we examine the characterisation of polyphenylenes and boron ($B_{12}$) using a line of symmetry. Our ability to quickly ascertain the valences of each atom, and the total number of atom bonds is made possible by the symmetrical chemical structures of polyphenylenes and boron $B_{12}$. By constructing these structures with degree-based indices, namely the K Banhatti indices, $ReZ{G}_1$-index, $ReZ{G}_2$-index, and $ReZ{G}_3$-index, we are able to determine their respective entropies.

In order to make quantitative structure-movement/property/danger relations, topological indices (TIs) are the numbers that are related to subatomic graphs. Some fundamental physicochemical properties of chemical compounds, such as breaking point, protection, and strain vitality, correspond to these TIs. In the compound graph hypothesis, the concept of TIs was developed in view of the degree of vertices. In investigating minimizing exercises of Star of David, these indices are useful. In this study, we explore the different types of Zagreb indices, Randić indices, atom-bond connectivity indices, redefined Zagreb indices, and geometric-arithmetic index for the Star of David. The edge partitions of this network are tabled based on the sum of degrees-of-end vertices and the sum of degree-based edges. To produce closed formulas for some degree-based network TIs, these edge partitions are employed.

In this research article, we motivate and introduce the concept of possibility belief interval-valued N-soft sets. It has a great significance for enhancing the performance of decision-making procedures in many theories of uncertainty. The N-soft set theory is arising as an effective mathematical tool for dealing with precision and uncertainties more than the soft set theory. In this regard, we extend the concept of belief interval-valued soft set to possibility belief interval-valued N-soft set (by accumulating possibility and belief interval with N-soft set), and we also explain its practical calculations. To this objective, we defined related theoretical notions, for example, belief interval-valued N-soft set, possibility belief interval-valued N-soft set, their algebraic operations, and examined some of their fundamental properties. Furthermore, we developed two algorithms by using max-AND and min-OR operations of possibility belief interval-valued N-soft set for decision-making problems and also justify its applicability with numerical examples.

In this paper, Dombi t-norm (TN) and Dombi t-conorm (TCN) are used to generate more complex, flexible and feasible operation rules by managing a parameter in bipolar neutrosophic fuzzy (BNF) environment. We introduce the notion of bipolar neutrosophic Dombi weighted geometric aggregation (BNDWGA) and bipolar neutrosophic Dombi ordered weighted geometric aggregation (BNDOWGA) operators. We discuss their different properties along with proofs and also investigate multi-attribute decision making (MADM) methods on basis of propose aggregation operators under bipolar neutrosophic fuzzy (BNF) environment. We propose an algorithm of selection of cultivating crop to explain the proposed methods under bipolar neutrosophic fuzzy (BNF) environment. The effect of parameter on our proposed bipolar neutrosophic Dombi aggregation operators are discussed graphically. Moreover, the comparison of our proposed methods with existing methods to test their suitable preferences are discussed in detail.