Abstract
In mathematical chemistry, the algebraic polynomial serves as essential for calculating the most accurate expressions of distance-based, degree-distance-based, and degree-based topological indices. The chemical reactivity of molecules, which includes their tendency to engage in particular chemical processes or go through particular reactions, can be predicted using topological indices. Considerable effort has been put into examining the many topological descriptors of simple graphs using ring structures and well-known groups instead of nonassociative algebras, quasigroups, and loops. Both finite quasigroups and loops are the generalizations of groups. In this article, we calculate topological descriptors and some well-known polynomials, -polynomial, Hosoya’s polynomial, Schultz’s polynomial, and modified Schultz polynomial of finite relatively prime graphs of most orders connected with two classes of quasigroups and go through their graphical aspects.
1. Introduction and Basic Definitions
A subfield of theoretical chemistry known as “chemical graph theory” studies chemical structures and reactions using the concepts of graph theory. A mathematical framework known as graph theory allows molecules to be represented as graphs, with chemical bonds acting as edges and atoms acting as vertices. The representational form utilized in mathematical models of chemical molecules is called a molecular graph. Many topological and structural properties of these molecules are investigated using concepts from graph theory. For instance, the degree and number of edges among the vertices of a chemical compound—a physical entity—can be used to predict the compound’s boiling point. Thus, it is evident that when a chemical problem is described mathematically, the topology of the molecule structure plays a critical role in defining the favorable properties of the matching molecular structure. Between 1975 and 2023, several academics employed algebraic structures, rings, and groups to address problems related to graph theory. The author showed that the maximal prime order of the nontrivial subgroup of the finite abelian group is the vertex independence number of the intersecting graph connected to the abelian group [1]. But eight years after the publication of this paper, several mathematicians proposed a novel idea for figuring out a finite simple graph’s vertex independence number using the vertex degrees [2]. In 1990, the author used the class of finite groups whose Cayley’s graphs are planar to characterize well-known groups, quasi-Frobenius groups, and linked components of finite simple graphs, whose nodes were the noncentral conjugacy classes of the group investigated by [3, 4]. A thorough analysis has been done on degree-based topological descriptors of two distinct graphynes and minimum transmission, depending on certain parameters, in two-mode networks [5, 6]. In addition, Zaman et al. and Mondal et al. [7, 8] have provided a complete computation of topological indices related with regression models and two particular ring structures; however, the researchers are still in the process of discovering certain well-known descriptors and polynomials associated with nonassociative algebras.
From here, we use for the undirected, simple, and finite graph, in which the set of edges is and the nodes are . The distance, a positive integer, between any two distinct vertices and of can be denoted by , and is the degree of any vertex in . Regarding degree-based topological indices, the -polynomial has a similar function in calculating closed expressions of multiple degree-based topological indices [9]. The following is the definition of a graph -polynomial associated with polynomial ring (see [10]): where is the overall number of edges such that . In this article, we use instead of . In reality, a topological index is a mapping from the set of real numbers to a class of isomorphic finite simple graphs [8]. For a graph , any degree-based topological index can be written as follows: where represents a function that has been specifically selected for potential chemical applications [11]. The result shown above can also be expressed as
Zagreb indices were first developed by Gutman and Trinajstić in 1972. The following defines the first Zagreb index according to [12]:
Here is how the second Zagreb index is described:
In 2015, Furtula and Gutman introduced the term forgotten topological index, which is expressed as follows [13]:
The second modified Zagreb index was introduced by Nikolić et al. in 2003, and it is defined as follows [14]:
Hu et al. put forward the generalized Randić index, which has been extensively researched in the chemistry and mathematics [15]. The following is a definition of the generalized Randić index:
. The definition of the inverse Randić index is as follows:
. A connected graph’s symmetric division deg index with the following definition was given by Vukicević in 2010 (see [16]).
In 1987, Fajtlowicz introduced the concept of a graph’s harmonic index, which is in [17]. It is described by
The following is how Balaban introduced the inverse sum indeg index in 1982 (see [18]):
Furtula et al. presented the augmented Zagreb index, which can be summed up as follows [19]:
For a graph , the distance-based Wiener index is defined as follows:
The Hosoya polynomial, with derivatives at 1 yield the Wiener index, is one basic polynomial in the domain of distance-based topological indices [20]. The following formula represents the Hosoya polynomial of a graph related with : where is the total number of node pairings in with a distance of between them and . The Wiener index is obtained as follows using the Hosoya polynomial’s first derivative at :
For the study of structure-property interactions of molecules, Randić gave the hyper-Wiener index in 1993, in [21]. It is a different distance-based index and is expressed as follows:
Hyper-Wiener index can also be obtained with the help of the Hosoya polynomial according to Cash et al. [22].
A topological index that combines distance and degree was first introduced by Schultz in 1989 and is known as the Schultz index [23]. Graph ’s Schultz index can be obtained as
Afterward, the modified Shultz index, a degree-distance-based index with the following definition, was published in 1997 by Klavžar and Gutman [24].
For a graph , the Schultz polynomial in integral domain can be calculated by
A graph ’s modified Schultz polynomial in ring is written as follows:
The following are the relationships that connect the Shultz, modified Shultz indices, and the related polynomials:
Under the assumption of three groupoids, , (), and (), and the identities
a mathematical system (, ·, \, and /) is known as a quasigroup (see [25]). A quasigroup which satisfies the identity law, and for unique , is called Loop. If is some power of prime number , then is called p-loop (see [26]).
2. Motivation and Applications
The Wiener index initiated the path of topological indices in 1947, modeling the paraffin’s temperatures at boiling point as follows [27]: where , , and are constants for a given isomeric group, is the boiling point, and and are the Wiener index and polarity number, respectively. The quantitative structure-property relationships between boiling temperatures and hyper-Wiener index were found in a range of cyclic and acyclic alkanes [28]. The first and second Zagreb indices were demonstrated to be effective in the estimation of the total -electron energy of molecule [29]. The linear combination of the forgotten topological index and the first Zagreb index yields a mathematical model of several physicochemical properties of alkanes with good accuracy [13]. They were proposed for the approximation of stretched carbon skeleton [12]. Randić observed the association between the Randić index and physicochemical parameters of alkane such as boiling temperature, enthalpy of formation, and surface area. Encoding molecular structure information with topological indices has a low processing cost and a high predictive potential. Additionally, these molecular descriptors provide information on easily recognized structural properties. The interaction between the algebraic and graph theoretical characteristics of the simple graph is the main area of study for graphs constructed from nonassociative finite algebra. Information in communication theory can be related to this. Therefore, it makes sense to calculate the finite quasigroups’ topological indices for relatively prime graph.
Theorem 1 (Lagrange’s theorem). Let be the finite loop and be any element of . Then, divides the order of .
Theorem 2 (fundamental theorem of arithmetic). Any positive integer can be written as a product of the powers of prime numbers.
Theorem 3 (see [30]). With the help of two finite groups , cyclic group of order 2, and , even order group of residue classes, the algebraic structure is a quasigroup, where is a positive integer. We can denote this class of quasigroups by .
Theorem 4 (see [31]). Let and be a cyclic group of order containing an element of order greater than 2 and two-element group of residue classes, respectively. Then, the algebraic structure is a quasigroup. We represent this class of quasigroups by .
The layout of this work consists of the following two sections: in the first section, we calculate topological indices of two classes given in [30, 31], and in the second section, there are some polynomials of relatively prime graphs associated with these quasigroups.
3. Topological Indices and Finite Quasigroups
Definition 5 (relatively prime graph). A finite simple graph is said to be relatively prime graph if and only if each element of is the vertex of and ; i.e., orders of two distinct elements of are relatively prime.
Example 1. The following Table 1 and Figure 1 indicate quasigroup of order 12 and its relatively prime graph, respectively.
Theorem 6. A relatively prime graph is star if and only if is p-loop.
Proof. Let be a star graph, since the order of the identity element of is one and it is relatively prime to the order of each nonidentity element of . Moreover, any two nonidentity elements are not adjacent in . It is only possible when order of loop is some power of prime number by Theorem 1. Other direction of the proof is just consequence of the Lagrange theorem. It completes the proof.
Theorem 7. A relatively prime graph is always connected.
Proof. Because the vertex associated with identity element is adjacent to each vertex so trivially, we can say relatively prime graph is connected.
Theorem 8. Let be the relatively prime graph associated with , where and is the positive integer greater than 1. Then, the degree-based topological indices are as follows: (1)(2)(3)(4)(5)(6)(7)=(8)(9)(10)
Proof. The following are the vertex and edge partitions of relatively prime graph with Equations (4)–(13) yield the required results. where , , and .
Theorem 9. Let be the relatively prime graph associated with , where and is the positive integer greater than 1. Then, the distance-based topological indices are as follows: (1)(2)
Proof. Let and be two distance-based subsets of defined by where cardinalities of and are and , respectively. It completes the proof with the help of Equations (14) and (17).
Theorem 10. Let be the relatively prime graph associated with , where and is the positive integer greater than 1. Then, the degree-distance-based topological indices are as follows: (1)(2)
Proof. Let and be two subsets of defined by where and . It completes the proof.
If and , then Figure 2 indicates relatively prime graph of quasigroup in .
Theorem 11. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, we have the following degree-based topological indices: (1)(2)(3).(4)(5)(6)(7)(8)(9)(10)
Proof. The following are the partitions of and : where , , , , , , , and . Similarly, we can prove the other results. It completes the proof.
To understand some of the following theorems, we introduce a mapping defined by
where and are the sets of geometric sequences and alternate triangular numbers, respectively, with .
Theorem 12. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, degree-distance-based topological indices are as follows: (1)(2)
Proof. Let be the subsets of defined by where , , , , , , , , and . The proof is complete by Equations (21) and (22).
Numerical values , , , , and represent the relatively prime graph (see Figure 3 to support Theorem 12).
Theorem 13. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, we have the following degree-based topological indices: (1)(2)(3)(4)(5)(6)(7)(8)(9)(10)
Proof. The following are the partitions of and : with cardinalities , , , , , , , , and .
Theorem 14. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, distance-based topological indices are as follows: (1)(2)
Proof. Let and be two distance-based subsets of defined by with and .
Theorem 15. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, degree-distance-based topological indices are as follows: (1)(2)
Proof. Let be the subsets of defined by where , , , , , , , , and .
Let be an element of the class with , , , and . Then, Figure 4 shows the relatively prime graph for Theorem 15.
4. Algebraic Approach of Topological Indices with Graphical Representations
Theorem 16. Let be the relatively prime graph associated with , where and is the positive integer greater than 1. Then, we have the following polynomials of one and two variables in the integral domains and : (1)(2)(3)(4)
Proof. It completes the required proof.
Example 2. If and , then the graphical representation (Figure 5) indicates the surface of -polynomial for quasigroup in .
Theorem 17. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, polynomials in and are as follows: (1), where(2), where(3), where(4), where
Proof. The partitions of Theorem 11, Corollary 18, and Theorem 12 with Equations (1), (15), (21), and (22) give the required results.
Corollary 18. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, distance-based topological indices are as follows: (1)(2)
Proof. Polynomials are studied to facilitate the calculations of topological indices. So Equations (16) and (17) give the required Wiener index and hyper-Wiener index , where the Hosoya polynomial is given in Theorem 17.
Example 3. Numerical values , , , , and represent the graph of -polynomial (see Figure 6).
Theorem 19. Let be the relatively prime graph associated with , where , is an odd prime, is a natural number, and is a positive integer. Then, some well-known elements of and are as follows: (1), where(2), where(3), where(4), where
Proof. The partitions of Theorems 13, 14, and 15 with Equations (1), (15), (21), and (22) constitute the required results.
Example 4. Let be an element of the class with , , , and . Then, Figure 7 is the surface of -polynomial.
Theorem 20. Let be the relatively prime graph associated with , where , and are odd primes, and are natural numbers, and is a positive integer. Then, -polynomial of ring is given by where , , , and .
Proof. The following are the partitions of and : where , , , , , , , , and .
Theorem 21. Let be the relatively prime graph associated with , where , and are odd primes, and are natural numbers, and is a positive integer. Then, the following is the Hosoya polynomial in ring : where and .
Proof. Let and be two distance-based subsets of defined by such that and .
Theorem 22. Let be the relatively prime graph associated with , where , and are odd primes, and are natural numbers, and is a positive integer. Then, the Schultz polynomial of integral domain is where
Proof. Let be the subsets of defined by with , , , , , , , , and .
Theorem 23. Let be the relatively prime graph associated with , where , and are odd primes, and are natural numbers, and is a positive integer. Then, the following is the modified Schultz polynomial of integral domain : where
Example 5. Let be an element of the class with , , , , , and . Then, Figures 8–10 give relatively prime graph, surface of -polynomial, and parabolic curve of modified Schultz polynomial, respectively.
5. Conclusion and Future Directions
This paper is the portrayal of multidisciplinary research among algebra, graph theory, and chemical graph theory. We have calculated degree, distance, and degree-distance-based topological indices of relatively prime graphs associated with two classes of quasigroups and . -polynomials, Hosoya polynomials, Schultz polynomials, and modified Schultz polynomials corresponding to these two classes indicate the three dimensional graphical representations known as surfaces , plane curves , , and , respectively. In other words, we can say that it is an effort to understand the topology of a particular graph related with nonassociative algebras through algebraic polynomials.
The isotopy-isomorphy property of these quasigroups allows us to call them G-loops, as most of their structural properties are very similar to those of finite groups. So the graphical study of flexible quasigroups, Bol quasigroups, and alternative quasigroups is still new to learn. In time to come, it will be very interesting to study adjacency and Laplacian spectrum, vertex connectivity, edge connectivity, and algebraic connectivity of these graphs interrelated with nonassociative binary operations and their connected applications like data structure and Cheeger’s inequality.
Data Availability
The work’s supporting data are referenced throughout the text.
Conflicts of Interest
There are no conflicts of interest, according to the authors.
Acknowledgments
The authors extend their appreciation to the Deanship of Scientific Research at Northern Border University, Arar, KSA, for funding this research work through the project number (NBU-FFR-2024-1519-01).