# On Characterization of Graphs Structures Connected with Some Algebraic Properties

Authors: Rongbing Huang; Muhammad Nadeem; Iqra Rashid; M.K. Siddiqui; Samuel Asefa Fufa
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, we have characterized graph structures connected with some algebraic properties. Also, this paper is actually the concatenation of graph theory and algebra. We have introduced left and right inverse graphs of antiautomorphic inverse property loops. Also, there is a connection between bipartite graphs and mathematical structures, commutator subloop, associator subloop, and associative part, the nucleus of the loop, through edge labeling.


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## Body

## 1. Introduction and Definitions

Ruth Moufang, a German geometer, invented the quasigroup, an extension of the nonassociative algebraic structure loop. In her Ph.D. thesis, she linked this algebraic structure to the nondesarguesian plane. Moufang was the first mathematician to propose nonassociative mathematical structures, following David Hilbert’s work on projective geometry [1]. She established that the octonion coordinates of the Cayley plane did not meet associative law using Desargues’ theorem. The Moufang loop is an algebraic structure that is connected to the Moufang plane. The tabular depiction of a loop or a quasigroup is known as a Latin square. We can solve issues in a variety of fields of mathematics using quasigroups and loops [2]. Mutually orthogonal quasigroups enable the creation of finite projective planes, which is one of the fundamental combinatorial problems associated with quasigroups, and algebraic nets are strongly related to loop structures in geometry via partial planes, projective planes, and affine planes [3, 4].In probabilistic reasoning of the experimental setup, blocking is the structuring of experimental material in clusters (blocks) that are analogous to one another. Blocking can be employed to solve the pseudoreplication problem. Row-column configurations for two limiting components in the research setup are known as Latin squares [5, 6]. An error correction code, in coding theory, information theory, telecommunications, and computing to control data errors over noisy communication or unreliable channels [7, 8] in cases where communication is disrupted by more types of sets of Latin squares that are orthogonal to each other have found application as error-correcting codes [9–11]. Determining if a substantially filled square can be completed to generate a Latin square is NP-complete [12].Through the use of combinatorics and group theory, Sudoku puzzles can be examined mathematically to solve several well-known problems. Latin squares can also be found in the more contemporary KenKen puzzles. Kamisado is a two-player abstract strategy board game played on a multicoloured board of order 8. Each player is in charge of an eight-piece octagonal dragon tower. Latin squares have been used in a number of board games, notably the widely used theoretical authentic puzzle game Kamisado [13]. In statistical sampling, a square grid comprising sample sites is considered as a Latin square if each row and column has only one sample [14]. A Latin hypercube is a multidimensional generalization of this concept, for each example being the only one in each coordinate axes hyperplane that incorporates it [15, 16]. According to graph theory, a rook’s graph is a mathematical structure that reflects all valid actions of the rook chess set on a chessboard. Each node of a rook’s graph indicates a face on a chessboard, but each edge depicts a permissible move from one face to the next. Two primary ways to describe these very same graphs mathematically are the Cartesian products of two simple graphs and the line graphs of bipartite graphs. In other words, we can say a graph with Latin squares as its colorings are known as Rook’s graph. In [17], the author proved that all three nuclei of antiautomorphic inverse property loops are coincident.The work on the seven bridges of Konigsberg, written by Leonhard Euler and published in 1736, is considered the very first article in graph theory’s history. This bough of mathematics plays an important role to solve many certifiable applications. Data structures are an area of computer science that employs graphs to represent data organization, computation flow, communication networks, and computing devices. Problems in social media, travel, biology, computer chip design, mapping the evolution of neurodegenerative disorders, and many other areas can all benefit from a similar approach (see [18, 19]). Additionally, the Feynman graphs and mathematical procedures embody particle physics concepts in a manner that is intimately related to the experimental values being investigated [20].Algebraic graph theory and group theory are strongly interconnected. The theory of algebraic graphs has been applied to a variety of fields, including dynamic systems and complexity. In molecular biology and genomics, graphs are frequently used to model and analyse datasets with complicated interactions. Graph-based techniques are extensively used in solitary transcriptomic research to group cells into cell types. Modeling the connections among genetic variants in a process, also including biosynthetic processes and genetic regulation systems, is another application [21]. Many graph theory problems and theorems have revolved around different techniques of colouring graphs. Four color problems are an example of this great effort. Modern coding theory makes substantial use of bipartite graphs, particularly to decipher coded language received from the network. This can be seen in Tanner graphs, and a set of vertices is partitioned into codeword digits and combinations of digits that should total to zero, and factor graphs, for example [22]. Petri nets make use of the qualities of directed bipartite graphs and other capabilities to allow mathematical demonstrations of system behaviour while also making simulations of the system simple to build [23]. According to the mathematical property that every two lines have at most one common point and every two points are collinear, Levi graphs need not have any cycles of length 4; thus, their girth has to be six or higher [24]. A bipartite graph’s biadjacency matrix is a matrix with 1 for each pair of nodes that are contiguous and 0 for nonadjacent nodes. Equivalences among hypergraphs, bipartite graphs, and directed graphs can be described using adjacency matrices.A hypergraph is a mathematical construction that has vertices and edges, very much like an undirected graph, but the edges can be any collection of vertices instead of needing to have exactly two ends. The bipartite graph can be considered as the directed graph’s bipartite double cover [25].A nonempty set⊙ is called groupoid or magma if the binary operation ⋄:⊙×⊙⟶⊙ is closed or internal. This algebraic structure can be denoted by ⊙,⋄. The unique solution of two equations ζ1⋄ζ2=ζ3 and ζ4⋄ζ1=ζ3 for ζ2,ζ4 implies magma ⊙,⋄ is a quasigroup ∀ζ1,ζ2,ζ3,ζ4∈⊙. In the presence of identity law, quasigroup can be taken as a loop with same left and right neutral element. Two self-bijections Jl:⊙⟶⊙ defined by ζJl=ζl and Jr:⟶⊙ defined by ζJr=ζr are known as left inverse permutation and right inverse permutation, respectively, of the loop ⊙ where ζl,ζr are said to be left inverse and right inverse of ζ, respectively, ∀ζ∈⊙. A loop ⊙ is called antiautomorphic inverse property loop (AAIPL) if and only if one of the following two equations are satisfied.(1)ζ1ζ2l=ζ2lζ1l,ζ1ζ2r=ζ2rζ1r,∀ζ1,ζ2∈⊙. Also, we define some very important algebraic substructures of ⊙. Sets(2)C⊙=ζ1,ζ2∈⊙|ζ1⋄ζ2=ζ2⋄ζ1⋄ζ1,ζ2∀ζ1,ζ2∈⊙,A⊙=ζ1,ζ2,ζ3∈⊙|ζ1⋄ζ2⋄ζ3=ζ1⋄ζ2⋄ζ3⋄ζ1,ζ2,ζ3∀ζ1,ζ2,ζ3∈⊙,Nl⊙=ζ1∈⊙|ζ1⋄ζ2⋄ζ3=ζ1⋄ζ2⋄ζ3∀ζ2,ζ3∈⊙,Nm⊙=ζ2∈⊙|ζ1⋄ζ2⋄ζ3=ζ1⋄ζ2⋄ζ3∀ζ1,ζ3∈⊙,Nr⊙=ζ3∈⊙|ζ1⋄ζ2⋄ζ3=ζ1⋄ζ2⋄ζ3∀ζ1,ζ2∈⊙,N⊙=Nl⊙∩Nm⊙∩Nr⊙,Z⊙=ζ1∈⊙|ζ1⋄ζ2=ζ2⋄ζ1∀ζ2∈⊙∩N⊙,are known as commutator subloop, associator subloop, left nucleus, middle nucleus, right nucleus, nucleus, and centre of the loop ⊙, respectively.AgraphΓ=Σ1,Σ2 is a concatenation of Σ2, an edge set, and Σ1, a vertex set. In a graph Γ, lines represent edges or links. Distance between two vertices ζ1 and ζ2 is the length of the shortest path from ζ1 to ζ2 denoted by dζ1,ζ2, and degree μζ of a vertex ζ is the number of shared edges [26, 27]. We can consider Γ as a finite graph if both sets Σ1 and Σ2 have finite number of elements. A graph, without loops and multiple edges, Γ is taken as a simple graph and an undirected graph if the edges do not indicate directions; otherwise, it can be taken as directed graph. If there is an edge between nodes ζ1 and ζ2 in set Σ2 of graph Γ, they are called neighbours (adjacent), and the edge is called incident in this case. In the undirected graph Γ, vertices of degree 0 and 1 are called isolated and pendant, respectively. If there is an edge between every two different vertices, a simple graph Kn on n vertices is taken to be complete graph. Due to the presence of an edge between any two distinct vertices, graph Γ is taken as connected graph. A connected graph, without multiple edges and loops, is known as a network. The graph Γ is said to be bipartite if there is a partition Σ′,Σ″ of Σ1 such that every edge in Γ connects a vertex in Σ′ and a vertex in Σ″. If there must be an edge between the vertices, a bipartite graph is said to be complete bipartite graph, and this graph is denoted by Km1,m2 where Σ′=m1, Σ″=m2. Graph Γ is considered to be star graph, Km1,1, if any one of the sets Σ′,Σ″ is singleton and balanced bipartite graph, Km1,m1, if m1=m2. A simple graph’s edge labeling is a mapping η:G⟶Σ2, from G, set of integers or symbols, to Σ2. Let ⊙ be any loop of finite order p with Latin square aijp×p and let X=X1,X2,…,Xp, Y=Y1,Y2,…,Yp be any two disjoint sets of symbols, then the mappings η1:X×Y⟶aijp×p defined by Xi,Yj↦aij, η2:aijp×p⟶Σ2 are known as P-edge labeling. Table 1 and Figure 1 represent P-edge labeling.Table 1
Latin square of order 4.
η1,η2Y1Y2Y3Y4X1abcdX2bcdaX3cdabX4dabcFigure 1
P-edge labeling.Now, let⊙ be any loop of finite-order p with Latin square ai,ajp×p, then the mapping η:ai,ajp×p⟶Σ2 is known as V-edge labeling (see [28, 29]) for more details. Table 2 and Figure 2 indicate V-edge labeling.Table 2
Loop of order 4.
(a, 0)(a, 1)(b, 0)(b, 1)(a, 1)(a, 0)(b, 1)(b, 0)(b, 0)(b, 1)(a, 1)(a, 0)(b, 1)(b, 0)(a, 0)(a, 1)Figure 2
V-edge labeling.Moreover, a directed graphΓ⊙,⋄ζ,ζlΓ⊙,⋄ζ,ζr is called a left inverse graph (right inverse graph) if Σ1=⊙ and Σ2=ζ,ζl:∀ζ∈⊙Σ2=ζ,ζr:∀ζ∈⊙ where the edge ζ,ζlζ,ζr indicates the initial vertex ζ and final vertex ζlζr in ⊙.


## 2. Main Results

LetΩ1, Ω2=ℤ2 be any nonabelian group and finite Boolean ring, respectively, with β1ϑ1,ϑ2,β2ϑ1,ϑ2,β3ϑ1,ϑ2,β4,β5,β6 be the nonempty set of mappings β1ϑ1,ϑ2:Ω1⟶Ω1, β2ϑ1,ϑ2:Ω1⟶Ω1, β3ϑ1,ϑ2:Ω1×Ω1⟶Ω1×Ω1, β4:Ω1⟶Ω1, β5:Ω1×Ω1⟶Ω1×Ω1, β6:Ω1×Ω1⟶Ω1 where ϖβ4=ϖ−1, inverse function, ϖ1,ϖ2β5=ϖ2,ϖ1, reverse function, ϖ1,ϖ2β6=ϖ2ϖ1∀ϖ,ϖ1,ϖ2∈Ω1, and ∀ϑ1,ϑ2∈Ω2.Theorem 1.
Let⋄ be the binary operation on the set ⊙=Ω1×Ω2 defined by(3)ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2,with(4)ϖ1∧ϑ1,ϑ2ϖ2=ϖ1β1ϑ1,ϑ2,ϖ2β2ϑ1,ϑ2β3ϑ1,ϑ2β6,where β1ϑ1,ϑ2=β4ϑ2,β2ϑ1,ϑ2=β4ϑ1,β3ϑ1,ϑ2=β5ϑ1ϑ2∀ϖ1,ϖ2∈Ω1, and ∀ϑ1,ϑ2∈Ω2. Then, ⊙,⋄ is the loop. Moreover, Γ⊙,⋄P=K2Δ,2Δ for the cardinality Δ of Ω1.Proof.
Clearly by the definition of⋄ algebraic structure ⊙,⋄ is groupoid or magma. And, ∀ϖ,ϑ∈⊙, then there exists 1,0∈⊙ such that(5)ϖ,ϑ⋄1,0=1,0⋄ϖ,ϑ=ϖ,ϑ,ϖ,ϑ⋄1,0=ϖ∧ϑ,01,ϑ+0,ϖβ40,1β4ϑβ50β6,ϑ=ϖ,1β6,ϑ=ϖ,ϑ.
And,(6)1,0⋄ϖ,ϑ=1∧0,ϑϖ,0+ϑ1β4ϑ,ϖβ40β50β6,ϑ1,ϖβ6,ϑϖ,ϑ.
Now,∀ϖ,ϑ∈⊙, then there exists ϖ−1,−ϑ∈⊙ such that(7)ϖ,ϑ⋄ϖ−1,−ϑ=ϖ−1,−ϑ⋄ϖ,ϑ=1,0,ϖ,ϑ⋄ϖ,ϑr=ϖ,ϑ⋄ϖ−1,−ϑ=ϖ∧ϑ,−ϑϖ−1,ϑ−ϑϖ,ϑ⋄ϖ,ϑr=ϖ∧ϑ,−ϑϖ−1,0.Case 1: Ifϑ=0,(8)ϖ∧ϑ,−ϑϖ−1=ϖβ4−ϑ,ϖ−1β4ϑβ5ϑ−ϑβ6=ϖ,ϖ−1β6=ϖ−1ϖ=1.Case 2: Ifϑ=1,(9)ϖ∧ϑ,−ϑϖ−1=ϖβ4−ϑ,ϖ−1β4ϑβ5β6=ϖ−1,ϖβ5β6=ϖ,ϖ−1β6=ϖ−1ϖ=1ϖ,ϑl⋄ϖ,ϑ=ϖ−1,−ϑ⋄ϖ,ϑ=ϖ−1∧−ϑ,ϑϖ,−ϑ+ϑ=ϖ−1∧−ϑ,ϑϖ,0ϖ,ϑl⋄ϖ,ϑ=ϖ−1∧−ϑ,ϑϖ,0.Case 3: Ifϑ=0,(10)ϖ−1∧−ϑ,ϑϖ=ϖ−1β4ϑ,ϖβ4−ϑβ5β6=ϖ−1,ϖβ6=ϖϖ−1=1.Case 4: Ifϑ=1,(11)ϖ−1∧−ϑ,ϑϖ=ϖ−1β4ϑ,ϖβ4−ϑβ5β6=ϖ,ϖ−1β5β6=ϖ−1,ϖβ6=ϖ.ϖ−1=1.
We have the result by using Case 1, Case 2 in (7) and Case 3, Case 4 in Equation 2. Since Ω1=Δ so ⊙,⋄=2Δ; thus, by P-edge labeling and without taking care of ⋄, the associated graph of ⊙,⋄ is K2Δ,2Δ.Theorem 2.
With the usual notations, set⊙,⋄ is an antiautomorphic inverse property loop. Also, Γ⊙,⋄ζ,ζl=Γ⊙,⋄ζ,ζr∀ζ∈⊙,⋄.Proof.
∀ϖ1,ϑ1,ϖ2,ϑ2∈⊙,⋄(12)ϖ1,ϑ1⋄ϖ2,ϑ2r=ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2r,ϖ1,ϑ1⋄ϖ2,ϑ2r=ϖ1β4ϑ2,ϖ2β4ϑ1β5ϑ1ϑ2β6,ϑ1+ϑ2r,ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2−1,−ϑ2⋄ϖ1−1,−ϑ1,ϖ2−1∧−ϑ2,−ϑ1ϖ1−1,−ϑ2−ϑ1ϖ2−1β4−ϑ1,ϖ1−1β4−ϑ2β5−ϑ2−ϑ1β6,−ϑ2−ϑ1ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2−1β4−ϑ1,ϖ1−1β4−ϑ2β5−ϑ2−ϑ1β6,−ϑ2−ϑ1.Case 1: Ifϑ1=0 and ϑ2=0,(13)ϖ1,ϑ1⋄ϖ2,ϑ2r=ϖ1,ϖ2β6,ϑ1+ϑ2r=ϖ2ϖ1,ϑ1+ϑ2r=ϖ1−1ϖ2−1,−ϑ1−ϑ2ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2−1,ϖ1−1β6,−ϑ2−ϑ1ϖ1−1ϖ2−1,−ϑ2−ϑ1.Case 2: Ifϑ1=1 and ϑ2=0,(14)ϖ1,ϑ1⋄ϖ2,ϑ2r=ϖ1,ϖ2−1β6,ϑ1+ϑ2r=ϖ2−1ϖ1,ϑ1+ϑ2r=ϖ1−1ϖ2,−ϑ1−ϑ2ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2−1,−ϑ2ϖ1−1,−ϑ1=ϖ2−1∧−ϑ2,−ϑ1ϖ1−1,−ϑ2−ϑ1=ϖ2−1β4−ϑ1,ϖ1−1β4−ϑ2β5−ϑ2−ϑ1β6,−ϑ2−ϑ1=ϖ2,ϖ1−1β6,−ϑ2−ϑ1=ϖ1−1ϖ2,−ϑ2−ϑ1.Case 3: Ifϑ1=0 and ϑ2=1,(15)ϖ1,ϑ1⋄ϖ2,ϑ2r=ϖ1−1,ϖ2β6,ϑ1+ϑ2r=ϖ2ϖ1−1,ϑ1+ϑ2r=ϖ1ϖ2−1,−ϑ1−ϑ2ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2−1,ϖ1β6,−ϑ2−ϑ1ϖ1ϖ2−1,−ϑ2−ϑ1.Case 4: Ifϑ1=1 and ϑ2=1,(16)ϖ1,ϑ1⋄ϖ2,ϑ2r=ϖ1−1,ϖ2−1β5β6,ϑ1+ϑ2r=ϖ1−1ϖ2−1,ϑ1+ϑ2r=ϖ2ϖ1,−ϑ1−ϑ2ϖ2,ϑ2r⋄ϖ1,ϑ1r=ϖ2,ϖ1β5β6,−ϑ2−ϑ1=ϖ2ϖ1,−ϑ2−ϑ1.
Therefore, the loop⊙,⋄ is AAIPL by Equations (12) and (13) and these four cases. Moreover, we have seen that the left and right inverse graphs of any arbitrary element ζ of ⊙,⋄ are the same. So, the left and right inverse graphs of this AAIPL are the same, Γ⊙,⋄ζ,ζl=Γ⊙,⋄ζ,ζr∀ζ∈⊙,⋄.Theorem 3.
Let⊙ be the AAIPL. Then, the set drΩ1×0 is a commutator subloop of ⊙. Moreover, ΓC⊙V=K1,Δ′ where Δ′=drΩ1 and drΩ1 are the derived subgroup of Ω1.Proof.
Letϖ,ϑ be the commutator of any two elements ϖ1,ϑ1 and ϖ2,ϑ2 in the loop ⊙,⋄. Case 1: Ifϑ1=0 and ϑ2=0,(17)ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ2,ϑ2⋄ϖ1,ϑ1⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ2∧ϑ2,ϑ1ϖ1,ϑ2+ϑ1⋄ϖ,ϑ,ϖ1β4ϑ2,ϖ2β4ϑ1β5ϑ1ϑ2β6,ϑ1+ϑ2=ϖ2β4ϑ1,ϖ1β4ϑ2β5ϑ2ϑ1β6,0⋄ϖ,ϑ,ϖ1,ϖ2β6,0=ϖ2,ϖ1β6,0⋄ϖ,ϑ,ϖ2ϖ1,0=ϖ1ϖ2,0⋄ϖ,ϑ,=ϖ1ϖ2∧0,ϑϖ,ϑ=ϖ1ϖ2β4ϑ,ϖβ40β50ϑβ6,ϑ=ϖ1ϖ2β4ϑ,ϖβ6,ϑ=ϖϖ1ϖ2β4ϑ,ϑ=ϖϖ1ϖ2,ϑϖ2ϖ1,0=ϖϖ1ϖ2,ϑ.Equation (17) is true only when ϖ=ϖ2ϖ1ϖ2−1ϖ1−1 and ϑ=0.Case 2: Ifϑ1=1 and ϑ2=0,(18)ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ2,ϑ2⋄ϖ1,ϑ1⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ2∧ϑ2,ϑ1ϖ1,ϑ2+ϑ1⋄ϖ,ϑ,ϖ1β4ϑ2,ϖ2β4ϑ1β5ϑ1ϑ2β6,1=ϖ2∧ϑ2,ϑ1ϖ1,1⋄ϖ,ϑ,ϖ1,ϖ2−1β6,1=ϖ2β4ϑ1,ϖ1β4ϑ2β5ϑ2ϑ1β6,1⋄ϖ,ϑ,ϖ2−1ϖ1,1=ϖ2−1,ϖ1β6,1⋄ϖ,ϑ=ϖ1ϖ2−1,1⋄ϖ,ϑ=ϖ1ϖ2−1∧1,ϑϖ,1+ϑ=ϖ1ϖ2−1β4ϑ,ϖβ41β51ϑβ6,1+ϑ=ϖ1ϖ2−1,ϖ−1β6,1+ϑ=ϖ−1ϖ1ϖ2−1,1+ϑϖ2−1ϖ1,1=ϖ−1ϖ1ϖ2−1,1+ϑ.Equation (18) holds when ϖ=ϖ1ϖ2−1ϖ1−1ϖ2 and ϑ=0.Case 3: Ifϑ1=0 and ϑ2=1,(19)ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ2,ϑ2⋄ϖ1,ϑ1⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ2∧ϑ2,ϑ1ϖ1,ϑ2+ϑ1⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2ϖ2,1=ϖ2β4ϑ1,ϖ1β4ϑ2ζϑ2ϑ1β6,1⋄ϖ,ϑ,ϖ1β4ϑ2,ϖ2β4ϑ1ζϑ1ϑ2β6,1=ϖ2,ϖ1−1β6,1⋄ϖ,ϑ,ϖ1−1,ϖ2β6,1=ϖ1−1ϖ2,1⋄ϖ,ϑ,ϖ2ϖ1−1,1=ϖ1−1ϖ2∧1,ϑϖ,1+ϑ=ϖ1−1ϖ2β4ϑ,ϖβ4ζϑβ6,1+ϑ=ϖ−1ϖ1−1ϖ2,1+ϑϖ2ϖ1−1,1=ϖ−1ϖ1−1ϖ2,1+ϑ.Equation (19) holds when ϖ=ϖ1−1ϖ2ϖ1ϖ2−1 and ϑ=0.Case 4: Ifϑ1=1 and ϑ2=1,(20)ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ2,ϑ2⋄ϖ1,ϑ1⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ2∧ϑ2,ϑ1ϖ1,ϑ2+ϑ1⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2ϖ2,0=ϖ2β4ϑ1,ϖ1β4ϑ2β5ϑ1ϑ2β6,0⋄ϖ,ϑ,ϖ1β4ϑ2,ϖ2β4ϑ1β5ϑ1ϑ2β6,0=ϖ2−1,ϖ1−1β5β6,0⋄ϖ,ϑ,ϖ1−1,ϖ2−1β5β6,0=ϖ2−1ϖ1−1,0⋄ϖ,ϑ,ϖ2−1,ϖ1−1β6,0=ϖ2−1ϖ1−1∧0,ϑϖ,ϑ,ϖ1−1ϖ2−1,0=ϖ2−1ϖ1−1β4ϑ,ϖβ40β50β6,ϑ,=ϖϖ2−1ϖ1−1,ϑϖ1−1ϖ2−1,0=ϖϖ2−1ϖ1−1,ϑ.
Equation (20) is true only when ϖ=ϖ1−1ϖ2−1ϖ1ϖ2 and ϑ=0. Thus, by Equations (19)–(22), the commutator subloop is drΩ1×0. And, the associated graph through V-edge labeling is K1,Δ′. □Theorem 4.
Let⊙ be the AAIPL. Then, the nontrivial set drΩ1×0 is the associator subloop of ⊙. Moreover, ΓA⊙V=K1,Δ′ where Δ′=drΩ1 and drΩ1 are the derived subgroup of Ω1.Proof.
Letϖ,ϑ be the associator of any three elements ϖ1,ϑ1, ϖ2,ϑ2, ϖ3,ϑ3 in the loop ⊙,⋄. Case 1: Ifϑ1=0, ϑ2=1, ϑ3=0,(21)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧ϑ2,ϑ3ϖ3,ϑ2+ϑ3=ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β40,ϖ3β41β5ϑ2ϑ3β6,1=ϖ1β4ϑ2,ϖ2β40β50.1β6,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2,ϖ3−1β6,1=ϖ2ϖ1−1,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3−1ϖ2,1=ϖ2ϖ1−1∧1,ϑ3ϖ3,1+ϑ3⋄ϖ,ϑ,ϖ1∧ϑ1,ϑ2+ϑ3ϖ3−1ϖ2,1=ϖ2ϖ1−1β40,ϖ3β41β51.0β6⋄ϖ,ϑ,ϖ1β41,ϖ3−1ϖ2β40β50.1β6,1=ϖ3−1ϖ2ϖ1−1,1⋄ϖ,ϑ,ϖ3−1ϖ2ϖ1−1,ϑ1+1=ϖ3−1ϖ2ϖ1−1∧1,ϑϖ,1+ϑ=ϖ3−1ϖ2ϖ1−1β40,ϖβ41β50.1β6,1=ϖ−1ϖ3−1ϖ2ϖ1−1,1ϖ3−1ϖ2ϖ1−1,1=ϖ−1ϖ3−1ϖ2ϖ1−1,1.Equation (21) holds when ϖ=1 and ϑ=0.Case 2: Ifϑ1=0, ϑ2=0, ϑ3=0,(22)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧ϑ2,ϑ3ϖ3,ϑ2+ϑ3=ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β40,ϖ3β40β6,0=ϖ1β40,ϖ2β40β6,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3,ϖ2,0=ϖ2ϖ1,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3ϖ2,0=ϖ2ϖ1∧0,ϑ3ϖ3,0+ϑ3⋄ϖ,ϑ,ϖ1∧0,0ϖ3ϖ2,ϑ1+0=ϖ2ϖ1β40,ϖ3β40β6,0⋄ϖ,ϑ,ϖ1β40,ϖ3ϖ2β40β6,0=ϖ3ϖ2ϖ1,0⋄ϖ,ϑ,ϖ3ϖ2ϖ1,0=ϖ3ϖ2ϖ1∧0,ϑϖ,0+ϑ,=ϖ3ϖ2ϖ1β40,ϖβ40β6,ϑ,=ϖϖ3ϖ2ϖ1,ϑ,ϖ3ϖ2ϖ1,0=ϖϖ3ϖ2ϖ1,ϑ.Equation (22) is valid if ϖ=1 and ϑ=0.Case 3: Ifϑ1=1, ϑ2=0, ϑ3=1,(23)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧ϑ2,ϑ3ϖ3,ϑ2+ϑ3=ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β41,ϖ3β40β6,1=ϖ1β40,ϖ2β41β6,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3ϖ2−1,1=ϖ2−1ϖ1,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3ϖ2−1,1=ϖ2−1ϖ1∧1,ϑ3ϖ3,1+ϑ3⋄ϖ,ϑ,ϖ1∧1,1ϖ3ϖ2−1,1+1=ϖ2−1ϖ1β41,ϖ3β41β51β6,0ϖ,ϑ,ϖ1β41,ϖ3ϖ2−1β41β1β6,0=ϖ1−1ϖ2ϖ3−1,0⋄ϖ,ϑ,ϖ1−1ϖ2ϖ3−1,0=ϖ1−1ϖ2ϖ3−1∧0,ϑϖ,0+ϑ=ϖ1−1ϖ2ϖ3−1β40,ϖβ40β6,ϑ=ϖϖ1−1ϖ2ϖ3−1,ϑ=ϖ1−1ϖ2ϖ3−1,0=ϖϖ1−1ϖ2ϖ3−1,ϑ.The (23) is true for ϖ=1 and ϑ=0.Case 4: Ifϑ1=1, ϑ2=0, ϑ3=0,(24)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧ϑ2,ϑ3ϖ3,ϑ2+ϑ3=ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β40,ϖ3β40β6,0=ϖ1β40,ϖ2β41β6,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3,ϖ2,1=ϖ2−1ϖ1,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3ϖ2,1=ϖ2−1ϖ1∧1,ϑ3ϖ3,1+ϑ3⋄ϖ,ϑ,ϖ1∧1,ϑ3ϖ3ϖ2,ϑ1+0=ϖ2−1ϖ1β40,ϖ3β41β6,1⋄ϖ,ϑ,ϖ1β40,ϖ3ϖ2β41β6,1=ϖ3−1ϖ2−1ϖ1,1⋄ϖ,ϑ,ϖ2−1ϖ3−1ϖ1,1=ϖ3−1ϖ2−1ϖ1∧1,ϑϖ,1+ϑ=ϖ3−1ϖ2−1ϖ1β40,ϖβ41β6,1ϖ2−1ϖ3−1ϖ1,1=ϖϖ3−1ϖ2−1ϖ1,1.Case 5: Ifϑ1=0, ϑ2=0, ϑ3=1,(25)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧0,1ϖ3,0+1=ϖ1∧0,0ϖ2,0+0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β41,ϖ3β40β6,1=ϖ1β40,ϖ2β40β6,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2−1,ϖ3β6,1=ϖ2ϖ1,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3ϖ2−1,1=ϖ2ϖ1∧0,1ϖ3,0+1⋄ϖ,ϑ,ϖ1∧ϑ1,1ϖ3ϖ2−1,ϑ1+1=ϖ2ϖ1β41,ϖ3β40β6,1⋄ϖ,ϑ,ϖ1β41,ϖ3ϖ2−1β40β6,1=ϖ3ϖ1−1ϖ2−1,1⋄ϖ,ϑ,ϖ1−1,ϖ3ϖ2−1β6,1=ϖ3ϖ1−1ϖ2−1∧1,ϑϖ,1+ϑ,ϖ3ϖ2−1ϖ1−1,1=ϖ3ϖ1−1ϖ2−1β40,ϖβ41,1β6=ϖ3ϖ1−1ϖ2−1,ϖ−1β6,1=ϖ−1ϖ3ϖ1−1ϖ2−1,1ϖ3ϖ2−1ϖ1−1,1=ϖ−1ϖ3ϖ1−1ϖ2−1,1.Case 6: Ifϑ1=1, ϑ2=1, ϑ3=1,(26)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧1,1ϖ3,1+1=ϖ1∧1,1ϖ2,1+1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β41,ϖ3β41β5β6,0=ϖ1β41,ϖ2β41β5β6,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2−1,ϖ3−1β5β6,0=ϖ1−1ϖ2−1,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2−1ϖ3−1,0=ϖ1−1ϖ2−1∧0,1ϖ3,0+1⋄ϖ,ϑ,ϖ1∧ϑ1,0ϖ2−1ϖ3−1,ϑ1+0=ϖ1−1ϖ2−1β41,ϖ3β40β6,1⋄ϖ,ϑ,ϖ1β40,ϖ2−1ϖ3−1β41β6,1=ϖ3ϖ2ϖ1,1⋄ϖ,ϑ,ϖ3ϖ2ϖ11=ϖ3ϖ2ϖ1∧1,ϑϖ,1+ϑ,ϖ3ϖ2ϖ1,1=ϖ3ϖ2ϖ1β40,ϖβ41β6,1+ϑ=ϖ3ϖ2ϖ1,ϖ−1β6,1+ϑ=ϖ−1ϖ3ϖ2ϖ1,1+ϑϖ3ϖ2ϖ1,1=ϖ−1ϖ3ϖ2ϖ1,1+ϑ.The (26) is true for ϖ=1 and ϑ=0.Case 7: Ifϑ1=1, ϑ2=1, ϑ3=0,(27)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧1,0ϖ3,1+0=ϖ1∧1,1ϖ2,1+1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β40,ϖ3β41β6,1=ϖ1β41,ϖ2β41β5β6,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2,ϖ3−1β6,1=ϖ1−1ϖ2−1,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ3−1ϖ2,1=ϖ1−1ϖ2−1∧0,0ϖ3,0+0⋄ϖ,ϑ,ϖ1∧ϑ1,0ϖ2−1ϖ3−1,ϑ1+1=ϖ1−1ϖ2−1β0,ϖ3β0β5,0⋄ϖ,ϑ,ϖ1β41,ϖ3−1ϖ2β41β5β6,0=ϖ3ϖ1−1ϖ2−1,0⋄ϖ,ϑ,ϖ1−1ϖ2−1ϖ3,0=ϖ3ϖ1−1ϖ2−1∧0,ϑϖ,0+ϑ,ϖ1−1ϖ2−1ϖ3,0=ϖ3ϖ1−1ϖ2−1β40,ϖβ40β6,ϑ=ϖ3ϖ1−1ϖ2−1,ϖβ6,ϑ=ϖϖ3ϖ1−1ϖ2−1,ϑϖ1−1ϖ2−1ϖ3,0=ϖϖ3ϖ1−1ϖ2−1,ϑ.Case 8: Ifϑ1=0, ϑ2=1, ϑ3=1,(28)ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3=ϖ1,ϑ1⋄ϖ2,ϑ2⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2∧1,1ϖ3,1+1=ϖ1∧0,1ϖ2,0+1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2β41,ϖ3β41β6,0=ϖ1β41,ϖ2β40β5β6,0⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2−1ϖ3−1,0=ϖ2ϖ1−1,1⋄ϖ3,ϑ3⋄ϖ,ϑ,ϖ1,ϑ1⋄ϖ2−1ϖ3−1,0=ϖ2ϖ1−1∧1,1ϖ3,0⋄ϖ,ϑ,ϖ1∧ϑ1,0ϖ2−1ϖ3−1,ϑ1+0=ϖ2ϖ1−1β1,ϖ3β1β5β6,0⋄ϖ,ϑ,ϖ1β40,ϖ2−1ϖ3−1β40β6,0=ϖ1ϖ2−1ϖ3−1,0⋄ϖ,ϑ,ϖ2−1ϖ3−1ϖ1,0=ϖ1ϖ2−1ϖ3−1∧0,ϑϖ,0+ϑ,ϖ2−1ϖ3−1ϖ1,0=ϖϖ1ϖ2−1ϖ3−1β40,ϖβ40β6,0,ϖ2−1ϖ3−1ϖ1,0=ϖϖ1ϖ2−1ϖ3−1,0.
Finally, by Equations (25)–(28), we have ϖ is any commutator of the nonabelian group Ω1 and ϑ=0. So, the subloop drΩ1×0 is the associator subloop of ⊙, and through V-edge labeling, ΓA⊙V=K1,Δ′ is the associated graph with this algebraic structure. □Theorem 5.
Let⊙ be the AAIPL. Then, set N⊙ is the trivial subloop of ⊙. And, ΓN⊙V=K1,1.Proof.
Letϖ,ϑ be in the left nucleus of the loop ⊙, then we can consider the following cases: Case 1: Ifϑ1=0, ϑ2=0,(29)ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2,ϖ,ϑ⋄ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ∧ϑ,ϑ1ϖ1,ϑ+ϑ1ϖ2,ϑ2,ϖ,ϑ⋄ϖ1ϖ2,ϑ1+ϑ2=ϖϖ1,ϑ+ϑ1⋄ϖ2,ϑ2,ϖ∧ϑ,ϑ1+ϑ2ϖ1ϖ2,ϑ+ϑ1+ϑ2=ϖϖ1∧ϑ+ϑ1,ϑ2ϖ2,ϑ+ϑ1+ϑ2,ϖ2ϖ1ϖ,ϑ+ϑ1+ϑ2=ϖ2ϖ1ϖ,ϑ+ϑ1+ϑ2.Case 2: Ifϑ1=0, ϑ2=1,(30)ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2,ϖ,ϑ⋄ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ∧ϑ,ϑ1ϖ1,ϑ+ϑ1ϖ2,ϑ2,ϖ,ϑ⋄ϖ1−1ϖ2,1=ϖϖ1,0⋄ϖ2,ϑ2,ϖ∧ϑ,1ϖ1−1ϖ2,ϑ+1=ϖϖ1∧0,1ϖ2,0+ϑ2,ϖ2ϖ1−1ϖ−1,1=ϖ2ϖ−1ϖ1−1,1.Case 3: Ifϑ1=1, ϑ2=0,(31)ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2,ϖ,ϑ⋄ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ∧ϑ,ϑ1ϖ1,ϑ+ϑ1ϖ2,ϑ2,ϖ,ϑ⋄ϖ1ϖ2,ϑ1+ϑ2=ϖϖ1,ϑ+ϑ1⋄ϖ2,ϑ2,ϖ∧ϑ,ϑ1+ϑ2ϖ1ϖ2,ϑ+ϑ1+ϑ2=ϖϖ1∧ϑ+ϑ1,ϑ2ϖ2,ϑ+ϑ1+ϑ2,ϖ2−1ϖ1ϖ−1,ϑ+ϑ1+ϑ2=ϖ2−1ϖ1ϖ−1,ϑ+ϑ1+ϑ2.Case 4: Ifϑ1=1, ϑ2=1,(32)ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2=ϖ,ϑ⋄ϖ1,ϑ1⋄ϖ2,ϑ2,ϖ,ϑ⋄ϖ1∧ϑ1,ϑ2ϖ2,ϑ1+ϑ2=ϖ∧ϑ,ϑ1ϖ1,ϑ+ϑ1ϖ2,ϑ2,ϖ,ϑ⋄ϖ1ϖ2,ϑ1+ϑ2=ϖϖ1,ϑ+ϑ1⋄ϖ2,ϑ2,ϖ∧ϑ,ϑ1+ϑ2ϖ1ϖ2,ϑ+ϑ1+ϑ2=ϖϖ1∧ϑ+ϑ1,ϑ2ϖ2,ϑ+ϑ1+ϑ2,ϖ1−1ϖ2−1ϖ,ϑ+ϑ1+ϑ2=ϖϖ1−1ϖ2−1,ϑ+ϑ1+ϑ2.
By Equations (29)–(32), we have a trivial subloop N⊙ of ⊙, and the associated graph through V-edge labeling is ΓN⊙V=K1,1.
LetΩ1=D8 and Ω2=ℤ2, then Figures 3–6 are the associated graphs of the loop, commutator subloop, associator subloop, the nucleus of the loop, and inverse graph of the loop, respectively, and Table 3 shows antiautomorphic inverse property loop.
LetΩ1 be the dihedral group of order 2n with drΩ1=1,ϖ,ϖ2,…,ϖn. Then, we can sum up all the above results in Table 4.Figure 3
Associated graph of a loopK16,16.Figure 4
Star graph of the substructureK1,4.Figure 5
Associated graph of the nucleusK1,1.Figure 6
Inverse graph of the loop.Table 3
AAIPL of order 16.
⋄(1, 0)(1, 1)(ϖ1,0)(ϖ1,1)ϖ12,0ϖ12,1ϖ13,0ϖ13,1(ϖ2, 0)(ϖ2, 1)(ϖ1ϖ2, 0)(ϖ1ϖ2, 1)ϖ12ϖ2,0ϖ12ϖ2,1ϖ13ϖ2,0ϖ13ϖ2,1(1, 0)(1, 0)(1, 1)(ϖ1,0)(ϖ1,1)ϖ12,0ϖ12,1ϖ13,0ϖ13,1(ϖ2, 0)(ϖ2, 1)(ϖ1ϖ2,0)(ϖ1ϖ2, 1)ϖ12ϖ2,0ϖ12ϖ2,1ϖ13ϖ2,0ϖ13ϖ2,1(1, 1)(1, 1)(1, 0)ϖ13,1ϖ13,0ϖ12,0ϖ12,1(ϖ1, 1)(ϖ1, 0)(ϖ2, 1)(ϖ2, 0)(ϖ1ϖ2, 1)(ϖ1ϖ2, 0)ϖ12ϖ2,1ϖ12ϖ2,0ϖ13ϖ2,0ϖ13ϖ2,0(ϖ1,0)(ϖ1,0)ϖ13,1ϖ12,0(1, 1)ϖ13,0(ϖ1,1)(1, 0)ϖ12,1ϖ13ϖ2,0(ϖ1ϖ2,1)(ϖ2,0)ϖ12ϖ2,1(ϖ1ϖ2,0)ϖ13ϖ2,1ϖ12ϖ2,0(ϖ2,1)(ϖ1,1)(ϖ1,1)ϖ13,0(1, 1)ϖ12,0ϖ13,1(ϖ1,0)ϖ12,1(1, 0)ϖ13ϖ2,1ϖ13ϖ2,0(ϖ2,1)(ϖ2,0)(ϖ1ϖ2,1)(ϖ1ϖ2,0)ϖ12ϖ2,1ϖ12ϖ2,0ϖ12,0ϖ12,0ϖ12,1ϖ13,0ϖ13,1(1, 0)(1, 1)(ϖ1,0)(ϖ1,1)ϖ12ϖ2,0ϖ12ϖ2,1ϖ13ϖ2,0ϖ13ϖ2,1(ϖ2,0)(ϖ2,1)(ϖ1ϖ2,0)(ϖ1ϖ2,1)ϖ12,1ϖ12,1ϖ12,1(ϖ1,1)(ϖ1,0)(1, 1)(1, 0)ϖ13,1ϖ13,0ϖ12ϖ2,1ϖ12ϖ2,0ϖ13ϖ2,1ϖ13ϖ2,0(ϖ2,1)(ϖ2,0)(ϖ1ϖ2,1)(ϖ1ϖ2,0)ϖ13,0ϖ13,0(ϖ1,1)(1, 0)ϖ12,1(ϖ1,0)ϖ13,1ϖ12,0(1, 1)(ϖ1ϖ2,0)ϖ13ϖ2,1ϖ12ϖ2,0(ϖ2,1)ϖ13ϖ2,0(ϖ1ϖ2,1)(ϖ2,0)ϖ12ϖ2,1ϖ13,1ϖ13,1(ϖ1,0)ϖ12,1(1, 0)(ϖ1,1)ϖ13,0(1, 1)ϖ12,0(ϖ1ϖ2,1)(ϖ1ϖ2,0)ϖ12ϖ2,1ϖ12ϖ2,0ϖ13ϖ2,1ϖ13ϖ2,0(ϖ2,1)(ϖ2,0)(ϖ2,0)(ϖ2,0)(ϖ2,1)(ϖ1ϖ2,0)(ϖ1ϖ2,1)ϖ12ϖ2,0ϖ12ϖ2,1ϖ13ϖ2,0ϖ13ϖ2,1(1, 0)(1, 1)(ϖ1,0)(ϖ1,1)ϖ12,0ϖ12,1ϖ13,0ϖ13,1(ϖ2,1)(ϖ2,1)(ϖ2,0)ϖ13ϖ2,0(ϖ1ϖ2,0)ϖ12ϖ2,1ϖ12ϖ2,0(ϖ1ϖ2,1)ϖ13ϖ2,0(1, 1)(1, 0)(ϖ1,1)ϖ13,1ϖ12,1ϖ12,0ϖ13,1(ϖ1,0)(ϖ1ϖ2,0)(ϖ1ϖ2,0)(ϖ1ϖ2,1)ϖ12ϖ2,0ϖ12ϖ2,1ϖ13ϖ2,0ϖ13ϖ2,1(ϖ2,0)(ϖ2,1)ϖ13,0ϖ13,1(1, 0)(1, 1)(ϖ1,0)(ϖ1,1)ϖ12,0ϖ12,1(ϖ1ϖ2,1)(ϖ1ϖ2,1)(ϖ1ϖ2,0)(ϖ2,1)ϖ12ϖ2,0ϖ13ϖ2,1ϖ13ϖ2,0ϖ12ϖ2,1(ϖ2,0)ϖ13,1(ϖ1,0)(1, 1)(1, 0)(ϖ1,1)ϖ13,0ϖ12,1ϖ12,0ϖ12ϖ2,0ϖ12ϖ2,0ϖ12ϖ2,1ϖ13ϖ2,0ϖ13ϖ2,1(ϖ2,0)(ϖ2,1)(ϖ1ϖ2,0)(ϖ1ϖ2,1)ϖ12,0ϖ12,1ϖ13,0ϖ13,1(1, 0)(1, 1)(ϖ1,0)(ϖ1,1)ϖ12ϖ2,1ϖ12ϖ2,1ϖ12ϖ2,(ϖ1ϖ2,1)ϖ13ϖ2,0(ϖ2,1)(ϖ2,0)ϖ13ϖ2,1(ϖ1ϖ2,0)ϖ12,1ϖ12,0ϖ13,1(ϖ1,0)(1, 1)(1, 0)(ϖ1,1)ϖ13,0ϖ13ϖ2,0ϖ13ϖ2,0ϖ13ϖ2,1(ϖ2,0)(ϖ2,1)(ϖ1ϖ2,0)(ϖ1ϖ2,1)ϖ12ϖ2,0ϖ12ϖ2,1(ϖ1,0)(ϖ1,1)ϖ12,0ϖ12,1ϖ13,0ϖ13,1(1, 0)(1, 1)ϖ13ϖ2,1ϖ13ϖ2,1ϖ13ϖ2,0ϖ12ϖ2,1(ϖ2,0)(ϖ1ϖ2,1)(ϖ1ϖ2,0)(ϖ2,1)ϖ12ϖ2,0(ϖ1,1)ϖ13,0ϖ12,1ϖ12,0ϖ13,1(ϖ1,0)(1, 1)(1, 0)Table 4
Graphs with algebraic structures.
Ω1AAIPLC⊙A⊙N⊙Γ⊙,⋄PΓC⊙VΓA⊙VΓN⊙VD4D4×ℤ2drD4×0drD4×01×0K8,8K2,1K2,1K1,1D6D6×ℤ2drD6×0drD6×01×0K12,12K3,1K3,1K1,1D8D8×ℤ2drD8×0drD8×01×0K16,16K4,1K4,1K1,1D2nD2n×ℤ2drD2n×0drD2n×01×0K4n,4nKn,1Kn,1K1,1


## 3. Conclusion and Future Applications

In this paper, we have shown, with the help of finite Boolean ring and nonabelian groups, the construction of nonassociative structures, antiautomorphic inverse property loops, and a graphical overview of loops, commutator subloop, nontrivial associator subloop, and trivial nucleus. It will be very interesting to see some other well-known mathematical structures that construct AAIPL and their associated graphs. This combination of algebra and graphs can be used in Simulink, and the adjacency matrices of left and right inverse graphs, which were presented here, can be employed in data structures. In computer systems that manage graphs, the adjacency matrix is a data structure and can be used to describe graphs. A commonly preferred data structure that is also utilised in this application is the adjacency list.



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*Source: 8792684-2022-06-08.xml*