Cayley Graphs over LA-Groups and LA-Polygroups

%e purpose of this paper is the study of simple graphs that are generalized Cayley graphs over LA-polygroups (GCLAP − graphs). In this regard, we construct two new extensions for building LA-polygroups. %en, we define Cayley graph over LA-group and GCLAP-graph. Further, we investigate a few properties of them to show that each simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAP-graph) is a GCLAP-graph and then we prove this result.


Introduction
e origins of graph theory can be traced back to Euler's work [1] on the königsberg bridges problem (1735), which thusly prompted the idea of an Eulerian graph. Graph is a mathematical portrayal of a grid and it portrays the relationship between lines and points. e idea of Cayley graph was introduced by Cayley [2] in 1878. Cayley graph has been widely studied in both directed and undirected forms. To study the characteristics of Cayley graphs, refer the papers [3][4][5][6].
First time Marty [7] introduced the concept of algebraic hyperstructures, which is a suitable extension of classical algebraic structures. Since then, a lot of works have been written on this topic. For a brief analysis of this theory, see [8,9]. In the books [10][11][12][13], we can see the applications of hyperstructures in lattices, cryptography, graph, automata, probability, geometry, and hypergraphs. A very good presentation of polygroup theory is in [14], which is utilized to consider color algebra [15][16][17] and hypergraph theory in [18] by Berge. e theory of left almost structures was first defined by Kazim and Naseeruddin [19] in 1972. Subsequently, Mushtaq and Kamran [20] established a new concept of left almost group (nonassociative group) called LA-group. e theory of left almost hyperstructures was first introduced by Hila and Dine [21] in the form of left almost semihypergroups, which was further investigated by Yaqoob et al. [22] and Amjad et al. [23]. In [24], Yaqoob et al. introduced the concept of LA-polygroups.
Recently, Heidari et al. [25] introduced a suitable generalization of Cayley graphs that is defined over polygroups (GCP − graphs) and showed that each simple graph of order ≥5 is a GCP-graph.
In this paper, we construct two new extensions for building LA-polygroups. en, we define the idea of Cayley graph over LA-group and GCLAP-graph. In particular, we proved some properties of them in order to show that each simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAPgraph) is a GCLAP-graph.

Preliminaries and Notations
is section contains some basic definitions of graph theory (see [26]) and left almost theory (see [24]).
A graph is represented by Φ � (R, D), where R is the set of vertices and D is the set of edges. Note that |R| is the order of a graph and |D| in a graph is its size. e graph K n is known as complete graph if every couple of vertices form an edge, where n is the number of vertices. In specific, K 1 is known as trivial graph and N n is known as null graph having no edges and n vertices.
where R ′ and R ″ are subsets of R and edges of the form d, f |d ∈ R ′ and f ∈ R ″ , then Φ � K m,n is known as complete bipartite graph, where |R ′ | � m and |R ″ | � n. In specific, K 1,n is known as star graph. e complement of a simple graph having no self edges and no multiple edges is known as simple graph.
Definition 1 (see [27]). If Φ � (R, D) is a graph and we form a sequence of vertices (ordered from left to right) d 1 , d 2 , d 3 , . . . , d n such that there is just one edge between every two succesive vertices and there are no other edges known as path. A path on n vertices is denoted by P n .
Definition 2 (see [27]). A graph is said to be a connected graph if there exists at least one path between every two vertices.
Definition 3 (see [27]). If all vertices have degree two of a connected graph, then it is called a cycle. A cycle graph has n vertices, represented by C n .
Definition 5 (see [24]). A multivalued system 〈L, ∘ , e, −1 〉, where e ∈ L −1 , is a unitary operation and ∘ maps L × L into the family of nonempty subsets of L which is called LApolygroup, if the following postulates hold for all d, f, w ∈ L: Example 2 (see [24]). Consider a finite set Q n � t 1 , t 2 , t 3 , . . . , t n , where n ≥ 3. en, Q n is an LA-polygroup under the following hyperoperation: For n � 5, we have the Cayley (Table 2).

(I) Extension of an LA-Polygroup by a Set
If exactly one of d, f, w ∈ L is equal to the left identity, then If exactly two of d, f, w ∈ L are equal to the left identity, then us, left invertive law holds. Now, we prove axiom (v) of Definition 5. Let d, f, w ∈ M such that w ∈ d ⊎f, then Case 1. If d, f, w ∈ L, then we have done.

Case 2. (a)
If w ∈ L * , then we have the following possibilities: us, condition (v) of Definition 5 holds and hence the theorem is proved. (Table 4).

(II) Extension of an LA-Polygroup by a Set
If d, f, w ⊈ L, then we consider the following cases: Case 3. If w � s k , f � s j , d � s i ∉ L in the following way: (i) such that k > j > i, then (ii) such that i � j, then (iii) such that j � k, then us, left invertive law holds. Now, we prove axiom (v) of Definition 5. Let d, f, w ∈ M such that w ∈ d⊎f. en, we consider the following cases: us, condition (v) of Definition 5 holds and hence the theorem is proved.  (Table 6). en, M 3 � L 3 4, 5 { } ◇ is an LA-polygroup with five elements and the Cayley (Table 7), where M 3 � 1, 2, 3, 4, 5 { }. is is an LA-polygroup and not a polygroup.

Cayley Graphs over LA-Groups and LA-Polygroups
Definition 6. Suppose that G is an LA-group and C is a subset of G such that  (i) 1 ∉ C, (ii) C − 1 � C; then, Cayley graph Cay(LAG, C) of G relative to C is the simple graph which has vertex set G and edge set D � g, gs |g ≠ gs, where g ∈ G and s ∈ C .
Example 6. e Cayley graphs of the LA-group G 5 � ( t 1 , t 2 , t 3 , t 4 , t 5 , * ) given in Example 1, with connection sets t 2 and t 2 , t 3 , are shown in Figures 1 and 2. For C � t 2 , we have For C � t 2 , t 3 , we have Definition 7. Given an LA-polygroup L � 〈L, ∘ , e, − I 〉 and L⊆C ≠ ∅ such that (C � C − I ), say the connection set. en, we define the generalized Cayley graph over LA-polygroup GCLAP(L; C) which is the simple graph having vertex set L and the edge set If we have an LA-polygroup L and a connection set C such that GCLAP(L; C) � Λ, then the graph Λ is known as a GCLAP-graph.
Here, we give few examples of GCLAP-graphs.

Example 7. e generalized Cayley graph of the left almost polygroup
{ } which is shown in Figure 3, where"°" is defined in Table 8.  Figure 4, where" ∘ " is defined in Table 9.

Which Simple Graphs Are GCLAP-Graphs?
First, we point out a few types of simple graphs that are GCLAP-graphs. After that, we infer that each simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAP-graph) is a GCLAPgraph.

Lemma 1. Every Cayley graph is a GCLAP-graph.
Proof. Since every LA-group is an LA-polygroup, therefore, by Definition 7, the result holds.

Lemma 2. Every complete graph of order at least three is a GCLAP-graph.
Proof. Let 〈Q n , * , t −1 1 〉 be an LA-polygroup, where n ≥ 3 (as defined in Example 2). en, GCLAP(Q n ; Q n ∖ t 1 ) are isomorphic to the complete graphs on n vertices, where n ≥ 3. Hence, it is proved. □ Table 6: LA-polygroup.

Mathematical Problems in Engineering
Lemma 3. Show that each star graph of order at least three is a GCLAP-graph.
Proof. Suppose that L � G n t n+1 ◇ , where n ≥ 3 and G n is defined in Example 1. en, Cayley table for L is given in Table 10. Now, by considering connection set C � t n+1 , we can see that S n � GCLAP(L; C). □ Lemma 4. If Φ is a GCLAP-graph. en, show that Φ ∪ nK 1 is also a GCLAP-graph, where n ≥ 1.
Proof. Let Φ be a GCLAP-graph. en, we have a left almost polygroup L and a connection set C such that Φ � GCLAP(L; C). Suppose that Φ n � Φ ∪ nK 1 and Z m � ∪ m t�1 z t . By Extension (II) and Definition 7, we have Φ 1 � GCLAP(L Z 1 ◇ ; C). Now, by using induction, Φ m � GCLAP(Q; C), where Q � L Z m ◇ and C is a connection set. Hence, Φ m+1 � GCLAP(Q z m+1 ◇ ; C). us, Φ n is a GCLAP − graph for every n ∈ N. Example 9. Pseudocomplete graphs on five vertices are shown in Figure 5.   e expansion of the graph Φ, represented by Φ + , is the join of the graph Φ and K 1 , i.e., Φ + � Φ∨K 1 .

Lemma 7. Show that the expansion of a GCLAP-graph is a GCLAP-graph.
Proof. Let Λ � GCLAP(L; C), where L is a left almost polygroup having n elements and C is a connection set. e faulty join graph, represented by ∇ g , is the graph such Moreover, a graph is known as ∇ 1 -graph if it is isomorphic to ∇ 1 (Φ, f), where Φ � GCLAP(L; C) such that L is a left almost polygroup having n elements, f ∉ L ,and C is a connection set.
Mathematical Problems in Engineering Proof. Suppose that Λ is a ∇ 1 -graph having n vertices. So, we have a left almost polygroup L having n elements and a connection set C such that Λ � ∇ 1 (GCLAP(L; C), n + 1). Let □ Hence, Λ is a GCLAP-graph. Up to here, we have determined a few types of GCLAPgraphs. Now, we confine ourselves to the graphs of order at most five (except cycle graph of order five) and show that every simple graph of order three, four, and five (except cycle graph of order five which may or may not be a GCLAPgraph) is a GCLAP-graph. In Appendix, we have shown all simple connected graphs of order three, four, and five (except cycle graph of order five) and denote them by Ω 1 , Ω 2 , . . . , Ω 28 . Theorem 3. All simple graphs of order three, four, and five (except cycle graph of order five which may or may not be a GCLAP-graph) are GCLAP-graphs.