A New Graphical Representation of the Old Algebraic Structure

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Introduction and Definitions
Te K € onigsberg problem was one of mathematics's most intriguing and illustrious puzzles in 1735. Tere was a wellknown Pregel River in the Russian city of K € onigsberg that had seven bridges. Four districts of the city were connected by these bridges. Te Russian president studied mathematics as well. He questioned whether one could explore every area of the community by crossing each bridge just once. He made several attempts to remedy this issue but was unsuccessful. Finally, he sent a letter to the Swiss mathematician Leonhard Euler asking for an explanation of the issue. Although Euler denied this letter and claimed it was not a work of mathematics, the query caught his interest [1]. He made a lot of efort.
In his study article from 1735, Euler explained the connection between this concept and others. His essay was 21 pages long. He thought that although this question has to do with geometry, this geometry is not the same as measurements and shapes [2]. Tis kind of geometry is novel. He created a sketch of this issue. He used dots for the terrain and lines for the bridges. Euler's path concept was then presented. Odd-degree nodes should be two or zero in number, and if there are two nodes of odd degree, they serve as the starting and ending points of a graph path that moves each edge just once. As a result, the sketch must follow the Euler path to provide the correct response to the earlier query. However, there are four odd-degree vertices in the given design. As a result, it deviates from Euler's path. As a result, there is no way to respond to the issue mentioned above, and the K€ onigsberg problem provided the basis for the frst graph theory theorem [3].
In the feld of mathematics known as algebraic graph theory, graph-related problems are solved using algebraic approaches [4]. Te area of algebraic graph theory known as group theory examines graphs in relation to the group theory. From 1975 to 2022, diferent researchers used algebraic structures, groups, rings, and loops, to overcome the problems connected with the graph theory. Te author demonstrated, with reference to [5], that the vertex independence number of the intersecting graph connected to this Abelian group is a maximal prime order of the nontrivial subgroup of the fnite Abelian group. He also provided this number for p groups. However, eight years after this article's publication, diferent mathematicians introduced a fresh concept for determining the vertex independence number of a fnite simple graph in terms of the vertex degrees [6].
Cayley's graphs for cyclic, symmetric, and dihedral groups, which are studied in the paper cited in [7], are another illustration of the link between algebra and graph theory. In 1990, the author used connected components of fnite simple graphs, whose vertices were the noncentral conjugacy classes of the group explored by [8] and the class of fnite groups whose Cayley's graphs are planar to characterise well-known groups, quasi-Frobenius groups [9]. Inspired work has been carried out in [10] to connect the total graph and its related induced subgraphs, the zerodivisor graph, the nilpotent elements graph, and the regular elements graph, which correspond to the commutative ring and its subsets, the sets of zero-divisors, the nilpotent elements, and the regular elements, and the substructures and comaximal ideals, of the ring associated with fnite graphs were introduced attractively in [11].
Te unit graph is a fnite simple graph in which any two diferent vertices have an edge if their total is the unit element of the ring and each node is an element of the fnite ring with nonzero identity. In [12], planarity, girth, diameter, chromatic index, and connectedness of unit graph were thoroughly explored. Mathematicians established novel ideas of cototal graph and counit graph, examined structural features of rings, and characterised a variety of rings after the algebraic and graphical properties connected to total graphs, unit graphs, and comaximal graphs [13]. To study graph-theoretic properties with ring-theoretic properties, it is important to examine the set of the nonzero zerodivisors of the commutative ring with identity. Te problems of when the zero-divisor graph of the commutative ring with identity is empty, fnite, and connected are quite reasonable. Te frst authors to respond to these inquiries in a general way can be seen in [14].
In addition, it would be interesting to know what conditions exist on the ring for the unit graph to become Hamiltonian. Te answer to this question was written by some authors with a citation to [15]. In-depth research was done on a very signifcant fnding, "unit and unitary Cayley's graph corresponding to a ring has genus 1 if and only if ring is the commutative Artinian" see [16]. Te prime graphs connected to solvable fnite groups provide as another illustration of algebraic graph theory.
Te authors introduced a new class of graphs called prime graphs in [17], where any two vertices x 1 and x 2 are the prime divisors of the order of fnite group and they are adjacent if the group has an element of order x 1 x 2 . In addition, they invented the notion of prime graphs for infnitely many solvable groups. During his work on colouring the graphs, the frst mathematician to propose the concept of zero-divisor graphs of the algebraic structure; commutative ring with nonzero identity was [18]. Diferent writers diligently investigated the fundamental characteristics of directed zero-divisor graphs connected to the ring of upper triangular matrices in 2013 by [19].
Te reference paper [20] also gave an essential direction for the future studies and connectedness of simple graphs, square elements graphs of fnite rings, was given in [21,22] attractively in the manner of very important characteristics of Cayley graphs, commuting graphs, intersection graphs, prime graphs, noncommuting graphs, conjugacy class graphs, and inverse graphs of fnite groups. Researchers in computer science and mathematics are currently interested in determining the energy of simple graphs, and the authors of [23] demonstrated some important fndings involving inverse graphs of two well-known kinds of fnite groups.
Upper bounds for the clique number and maximum degree associated with square graphs are given in [24,25], where authors also computed the result regarding the number of edges of this graph. Some authors calculated some results of equal-square graphs, a subclass of simple graphs where all the elements of the fnite group are the vertices and any two distinct vertices x 1 , x 2 are adjacent if x 2 1 � x 2 2 see [26]. With the aid of loop structures and a fnite Boolean commutative ring, the authors of [27,28] discovered properties of balanced bipartite graphs and zero-divisor graphs, while [29] gave a concept of directed inverse graphs of antiautomorphic inverse property loops and star graphs of substructures of these loops through edge labelings.
In order to clarify key characteristics of cycles, writers combined three well-known branches of mathematics: algebra, linear algebra, and graph theory. Tey also looked into a class of Koszul algebra using these cycles [30]. Interrelationship of complicated algebraic objects, burnside groups, and collectives of automata is nicely done in [31], and the study of chromatic numbers, clique numbers, cycles, and distance graphs presented comprehensively in the papers [32,33].
Chemical graph theory is a fascinating area of mathematics that combines graph theory and chemistry. Trough mathematical methods, the molecules from the Chemistry model produce a molecular graph. Atoms serve as the vertices and chemical bonds serve as the edges of a molecular graph. To get their diferent topological and structural characteristics, these molecular formations have been subjected to a variety of graph theory techniques [34]. For instance, the degree and distance between the vertices of a chemical compound can be used to determine the boiling point of the chemical compound. As a result, mathematicians might state that when describing a chemical problem when it comes of mathematical form, the molecular structure's topology is important in deciding the usable characteristics of the related chemical compound [35].
In 1988, it was estimated that a few hundred professional researchers produced 500 study publications annually researching diferent chemical structural aspects, including meticulously researched contents see [36]. Commercial, pharmaceutical, and industrial chemistry make extensive use of a number of chemical compounds possesses distinct mathematical structures. Atomic arrangements inside chemical compounds follow clear structural laws that have practical hidden properties. Chemical graph theory is a predominant bough of graph theory in practical research that explores these aspects using combinatorics and topology as mathematical tools. It is important to note that chemical graph theory has benefted much from mathematical chemistry [37][38][39]. Many of the invariants found in chemical graph theory, like indices or descriptors, are used in a variety of commercial areas, most notably in the chemical and pharmaceutical industries [40,41].
More specifcally, the development of associated felds is actively infuenced by the study of degree-based indices. For further study in this direction, one may read the papers bounds on the partition dimension of convex polytopes, on the partition dimension of trihexagonal alpha-boron nanotube, the locating number of hexagonal Mobius ladder network, and lower bounds for Gaussian Estrada index of graphs [42,43]. It is desirable to gather a lot of information in the way of numerical values connected to chemical structures using modern computer systems and to compare them [44]. In order to satisfy the demands of chemists, many topological descriptors were developed in the fnal ten years of the nineteenth century [45,46].
With the help of two foremost mathematical objects * :  (1) and (2): can be taken as identity law and in the presence of neutral element e of Q (FW) any quasigroup (Q (FW) , * ) or simply Q (FW) with this law is said to be a loop. Te left and right inverse of each element in a group are always the same; however, this is not true for loops. Self bijections are called left inverse permutation and right inverse permutation, respectively. Moreover, in addition of the following identities, equations (3) and (4) are as follows: x and x has a unique left inverse x λ . We denote the set of these elements of Q (FW) by I λ and it is straightforward to say that I λ ⊂ Q (FW) . In the similar way, we can defne I ρ . For the left inverse graph G I λ Q (FW) , a fnite simple graph, we assign all the elements of Q (FW) as vertices and any two distinct vertices x 1 , x 2 of G I λ Q (FW) are adjacent if and only if either x 1 * x 2 ∈ I λ or x 2 * x 1 ∈ I λ and analogously we can defne G I ρ Q (FW) . Let G I λ Q (FW) be the connected graph then the distance d(x 1 , x 2 ) between two diferent vertices x 1 , x 2 is the shortest path's length from x 1 to x 2 in G I λ Q (FW) and the number of edges that a vertex x shares is referred to as its degree d x .
Also, ecc( is called eccentricity of vertex x 1 , where minimum and maximum eccentricity are taken as radius and diameter of G I λ Q (FW) denoted by rad(G I λ Q (FW) ) and diam(G I λ Q (FW) ), respectively. According to Gutman and Trinajstc � [47,48], the Zagreb indices are as follows: Following are Došlić's defnitions of the Zagreb coindices [49]: In [50], Ghorbani and Azimi defne the multiple Zagreb indices as follows: In [51], Albertson initiated the irregularity of G as follows: Two other most of the time utilized topological descriptors that give the measurement about irregularity of a graph are the Collatz-Sinogowitz index and the degreebased variance index see [52]. Also, where λ 1 be the greatest eigenvalue of A � (a ij ), adjacency matrix, of the graph G.
Proof. Because the order of the fexible loop is 4 ∝ and it has only two loop-involutions by the previous theorem so the cardinality of □ It is natural to see that under what condition this inverse graph will be an empty graph. Following result gives the answer of this question.
□ Now, we see the following result to get rid of the sign < from the previous Teorem 10.
{ } with even ∝ � 2 then the following Table 1 and Figure 1 indicate fexible weak inverse property loop of order 8 and its inverse graph, respectively. Moreover, blue and red vertices in the Figure 1 represent self-loop-involutions and nonself-loopinvolutions of (C 2 × Z 4 , * ), respectively. Table 2 and Figure 2 indicate fexible weak inverse property loop of order 12 and its inverse graph, respectively. Moreover, blue and red vertices in the Figure 2 represent self-loop-involutions and nonself-loopinvolutions of (C 2 × Z 6 , * ), respectively.
Let 0 ≠ e ′ , o ′ be the even and odd numbers of Z 2∝ , respectively, with any element y p of C 2 , then Table 3 indicates the recapitulation of all the abovementioned results:

Topological Indices of Inverse Graph
In this section, we shall denote the inverse graph G Journal of Mathematics    5  6  6  3  12  10  9  9  10  10  4  16  14  13  13  14  14  5  20  18  17 17 18 18 Proof. With the help of equation (5), the frst Zagreb index can be obtained as follows: By equation (6), we can calculate second Zagreb index as follows: It completes the proof. □ Theorem 1 (see [54]). Let G � (V, E) be a simple graph then frst Zagreb coindex Z 1 (G) is given by the following equation: Theorem 16 (see [54]). Let G � (V, E) be a simple graph then second Zagreb coindex Z 2 (G) is given by the following equation: Theorem 17. Let Γ be the inverse graph of (C 2 × Z 2∝ , * ) then frst and second Zagreb coindices of Γ are Proof. Te frst Zagreb coindex by Teorem 15 and equation (7) can be written as follows: Also, with the help of Teorem 16 and equation (8) second Zagreb coindex can be calculated as follows: □ and Proof. Equation (9) helps us to fnd frst multiple Zagreb index as follows: By equation (10) the second multiple Zagreb index is given by the following equation: It completes the proof. □ Theorem 19. Let Γ be the inverse graph of (C 2 × Z 2∝ , * ) then irregularity index of Γ is irr(Γ) � 5 ∝ − 6.
Proof. By equation (13), the Collatz-Singowitz index can be calculated as follows: It completes the proof.
□ Table 4 indicates the values of all abovementioned indices.
As we can see the Figures 1 and 2, if ∝ increases then the order of inverse graphs obviously increases. Te important thing is only variance and CS indices decreases, they treat like decreasing functions but all the other indices move in the direction of ∝ . Moreover, in comparison, we have observed both frst and second Zagreb multiple indices increases exponentially. It means, mathematically we can write for the very very large ∝ these indices tend to infnity and on the other hand irregularity index shows a slow behavior but never overlap to zero because inverse graph is not regular.

Conclusion and Future Direction
In algebraic graph theory, we solve the problems related with graphs by taking into account some algebraic structures. Tis paper is actually a portrayal of this notion. Trough fexible loops, a generalisation of commutative loops, and inverse graphs, we have uncovered diferent important numbers of simple graphs. In future, we can emphasize the notions, energy, chromatic number, clique number, vertex connectivity, edge connectivity, algebraic connectivity, adjacency spectrum, Laplacian spectrum of inverse graphs associated with this algebraic structure, or other structures of nonassociative binary operations. Moreover we can discuss optimization problems of this class of simple graphs related with spectral and polynomial methods. Other algebraic fndings connected to adjacency matrices of inverse graphs of loop structures and their applications in data structure via adjacency list would be intriguing to study. In time to come, those molecular graphs that intersect with simple graphs discovered using mathematical structures will be engrossing and we shall be able to fnd Hosoya polynomial, Schultz polynomial, modifed Schultz polynomial, and Mpolynomial.

Data Availability
No data were used to support the study.

Conflicts of Interest
Te authors declare that they have no conficts of interest.