Algebraic Structure Graphs over the Commutative Ring Z m : Exploring Topological Indices and Entropies Using M -Polynomials

: 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 (cid:98) G is fully made up of the zero divisors of the modular ring Z n , the graph is said to be a zero-divisor graph. If the products of two vertices are equal to zero under (mod n ), 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.


Introduction
Topological indices and algebraic graph theory are two closely linked subjects that focus on the mathematical study of graphs, having applications in chemistry, physics, computer science and social networks.Topological indices and algebraic graph theory are linked by a shared interest in graph analysis and representation.Although topological indices are a specific set of numerical measurements obtained from graph topology, algebraic graph theory gives mathematical tools and notions for studying graph features, which can be used to analysis and understand topological indices and vice versa.
A molecular graph is a type of topological representation of a molecule that represents the structure and connections of a molecule.These molecular graphs characterize numerous chemical aspects of molecules, such as their organic, chemical, or physical properties.They are critical in applications such as quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) research, digital screens, and computational drug design [1,2].Many topological indices have been used to characterize molecular graphs and many of these indices are good graph descriptors [3,4].Furthermore, several of these indices have been discovered to correspond well with the organic, chemical, or physical characteristics of molecules [5][6][7][8][9][10][11][12][13][14][15][16][17].As a result, they serve an important role in understanding and predicting molecular behavior and characteristics in a variety of chemical and pharmacological situations.
Graphs in mathematics are made up of vertices (which represent atoms) and edges (which represent chemical bonds).A molecular graph is a graph that represents the structure and connectivity of molecules and acts as a topological representation of the molecule.These molecular graphs are analyzed using a variety of topological indices, including distance-based topological indices, degree-based topological indices, and other derived indices.Distance-based topological indices, in particular, have an important role in chemical graph theory, notably in chemistry [18,19].Each type of topological index offers unique information about the molecular graph and many have been presented to study different aspects of chemical compounds.Topological indices help in the analysis of molecular structures, property prediction, drug design, and other areas of chemical research.
Degree-based topological indices have undergone extensive research and have shown significant connections to various properties of the molecular compounds under study.The relationship between these indices is remarkably strong.Among the topological indexes derived from distance and degree in [20], degree-based topological indices stand out as the most widely recognized examples of such invariants.Numerical values exist that establish connections between the molecular structure and various physical properties, chemical reactivities and biological activities.These numerical values, known as topological indices, associate the molecular shape with specific physical properties, artificial reactivities, and natural biological activities [21,22].
In 1948, Shannon introduced the concept of entropy through his seminal paper [23].Entropy, when applied to a probability distribution, serves as a measure of the predictability of information content or the uncertainty of a system.Subsequently, the application of entropy extended to graphs and chemical networks, enabling a deeper understanding of their structural information.Graph entropies have recently gained significance in various domains, including biology, chemistry, ecology, sociology, among others.The idea of entropy, which derives from statistical mechanics and information theory, quantifies how random or unpredictable a system is.Even though entropy is most frequently related to information theory, it may also be used to examine complex systems, networks, and patterns in algebraic structures and graphs.A detailed survey on the application of algebraic entropies over algebraic structure has been presented in [24].The degree of each atom holds paramount importance, leading to substantial research in graph theory and network theory to explore invariants that have long served as information functionals in scientific studies.The chronological sequence of graph entropy measurements utilized to analyze biological and chemical networks [25][26][27].Further, Das et al. investigated various results on topological indices and entropies in their research articles [28][29][30][31][32][33].
A graph that depends on algebraic structures such as group theory, number theory, and ring theory is known as an algebraic graph.In the field of algebraic graph theory, various problems are still open; the number of components problem of a graph, which depends on the modular relation, still remains a conjuncture.There are several algebraic graphs based on the algebraic structure that has been studied, but here we are going to discuss a zero-divisor graph that depends on the set of zero divisors of a ring R. A graph G is known as a zero-divisor graph whose vertex set is the zero divisors of the modular ring Z n , and two vertices will be adjacent to each other if their product will be zero under (mod n) [34].The zero-divisor graph for Z 18 is shown in Figure 1.For further understanding related to algebraic structure graphs and their properties, reader should can study [35][36][37][38][39][40].The rest of the work is arranged as follows: In Section 2, some basic terminology related to topological indices are given to understand the proposed work.In Sections 3 and 4, various topological indices over zero-divisor graphs G(Z ℘ 2 1 ℘ 2 ) and G(Z ℘ 3 1 ℘ 2 ) are discussed, respectively.Further, in these sections the behavior of investigated topological indices via numeric tables and three-dimensional discrete plotting are observed numerically and graphically.In Section 5, three kinds of entropies, i.e., first, second, and third redefined entropies, are founded over the families of zero-divisor graphs.In the last section, concluding remarks and further future works are discussed with detail.
In chemical graph theory, the M-polynomial is a topological index used to characterize the molecular structure of organic molecules.Researchers can acquire insights into the structural factors that determine the properties of molecules by analyzing the coefficients of different terms in the M-polynomial.The M-polynomial is defined as [3,8] The relationship between M-polynomial and topological indices are given in Table 1.
Table 1.Relation between M-polynomial and topological indices.

Topological Indices
where, In 2013, Ranjini et al. introduced the first, second, and third redefined version of the Zagreb indices [50], The concept of entropy was introduced by Chen et al. in 2014 [51], and is defined as The remaining entropies were found in [52], which are defined as 1.
First redefined Zagreb entropy: if Then Now, by using ( 11) in (10), the first redefined Zagreb entropy is 2.
Second redefined Zagreb entropy: if Then Now, by using ( 13) in (10), the second redefined Zagreb entropy is Third redefined Zagreb entropy: if Then Now, by using ( 15) in (10), the third redefined Zagreb entropy is 3. M-Polynomial and Topological Indices for Zero-Divisor Graph G(Z ℘ 2 1 ℘ 2 ) Let Z n be a modular ring with unity and Z n × Z m be the product of two modular rings.A non-zero element z of a modular ring Z n is said to be a zero divisor if there exists another non-zero element y of Z n such that the product of z and y will be zero under the modulo n.In other words, two non-zero elements will be zero divisors to each other if their product will be zero.Similarly, two non-zero elements (x 1 , y 1 ), (x 2 , y 2 ) from Z n × Z m will be zero, divisors to each other if the product of both will be zero such that (x 1 , y 1 ) × (x 2 , y 2 ) = (0, 0) under modulo mn.In this section, M− polynomial and topological indices for the zerodivisor graph G(Z ℘ 2 1 ℘ 2 ) with numerically and graphically behavior are discussed.The zero-divisor graph G(Z 5 2 ×3 ) as shown in Figure 2.
Proof.Let G(Z p 2 ℘ 2 ) be a zero-divisor graph over Z p 2 ℘ 2 with distinct primes (p > ℘ 2 ).By adding ( 24) and ( 25); Differentiating with respect to y of Equation ( 41); Differentiating with respect to y of Equation ( 42); By substituting x = y = 1 in Equation ( 43) the third symmetric division index becomes; After some more simplification, we have By applying D x on Equation (28); Similarly; Adding Equations ( 46) and (47); By substituting x = y = 1 in Equation ( 48) the fifth symmetric division index becomes; After some more simplification, we have the desired result.
For the harmonic index, we will first find the value of J; Applying δ y on (50), we have The harmonic index is, For the inverse sum index, firsy applying J on (26); (53) Applying δ x on (53), we have The inverse sum index is, For the forgotten topological index, we will first compute D 2 x and D 2 y .Applying D x on (24); By adding Equations ( 55) and ( 56).
From Table 4 and Figure 6, we conclude that Further, different kinds of entropies such as the first, second, and third redefined Zagreb are investigated over proposed families of graphs.In future work, if anyone can generalize this study for each zero-divisor graph, then this result is very interesting for researchers working in the area of algebraic graph theory.

Figure 4 .
Figure 4.The three-dimensional discrete plot of third symmetric division, fifth symmetric division, harmonic, inverse sum and forgotten topological index for G(Z ℘ 2 1 ℘ 2 ).

4 .
M-Polynomial and Topological Indices for Zero-Divisor Graph G(Z ℘ 3 1 ℘ 2 ) In this section, M− polynomial and topological indices for zero-divisor graph G(Z ℘ 3 1 ℘ 2 ) with numerically and graphically behavior are discussed.The zero-divisor graph G(Z 3 3 ×2 ) as shown in Figure 5.