Abstract
The purpose of this paper is the study of simple graphs that are generalized Cayley graphs over LA-polygroups . In this regard, we construct two new extensions for building LA-polygroups. Then, 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.
1. Introduction
The origins of graph theory can be traced back to Euler’s work [1] on the königsberg bridges problem , 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.
The 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 [3–6].
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 [10–13], 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 [15–17] and hypergraph theory in [18] by Berge.
The 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. The 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 and showed that each simple graph of order is a GCP-graph.
In this paper, we construct two new extensions for building LA-polygroups. Then, 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 GCLAP-graph) is a GCLAP-graph.
2. Preliminaries and Notations
This section contains some basic definitions of graph theory (see [26]) and left almost theory (see [24]).
A graph is represented by , where is the set of vertices and is the set of edges. Note that is the order of a graph and in a graph is its size. The graph is known as complete graph if every couple of vertices form an edge, where is the number of vertices. In specific, is known as trivial graph and is known as null graph having no edges and n vertices.
If is a graph such that , where and are subsets of and edges of the form , then is known as complete bipartite graph, where and . In specific, is known as star graph. The complement of a simple graph is denoted by , where such that . Let and be two graphs. Then, is known as subgraph of , if and . A subgraph is known as induced subgraph, if contains each and every edge with . Two graphs and are said to be isomorphic, if , a bijection, such that . We denote this by . Let and be two graphs. Then, , where and and joint of two graphs with and . A graph having no self edges and no multiple edges is known as simple graph.
Definition 1 (see [27]). If is a graph and we form a sequence of vertices (ordered from left to right) 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 .
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 vertices, represented by .
Definition 4 (see [20]). A groupoid is called a left almost group, i.e., LA-group, if(i) such that for all ,(ii), such that ,(iii), .
Example 1 (see [20]). Let where , under binary operation which is defined asThen, is an LA-group. For , we have the Cayley (Table 1).
Definition 5 (see [24]). A multivalued system , where , is a unitary operation and maps into the family of nonempty subsets of which is called LA-polygroup, if the following postulates hold for all :(i)Left invertive law: ,(ii)Reproducibility axiom: ,(iii) is a left identity such that ,(iv),(v).
Example 2 (see [24]). Consider a finite set , where . Then, is an LA-polygroup under the following hyperoperation:For , we have the Cayley (Table 2).
Example 3 (see [24]). Consider with the hyperoperation , given in Table 3.
Then, is an LA-polygroup.
Notations. All LA-polygroups are represented by and the underlying sets are represented by Also, and . Now, we establish two new extensions for making LA-polygroups.
2.1. Extension of an LA-Polygroup by a Set:
Suppose that is an LA-polygroup and , where is a nonempty set. Put , , for every and ; we define
The system is known as the extension of LA-polygroup by a set and represented by .
Theorem 1. Let be an LA-polygroup and be a nonempty set such that . Then, is an LA-polygroup.
Proof. Suppose that , , , and . If , thenIf exactly one of is equal to the left identity, thenIf exactly two of are equal to the left identity, thenIf , then . Thus, left invertive law holds. Now, we prove axiom () of Definition 5. Let such that , then Case 1. If , then we have done. Case 2. If , then we have the following possibilities:(i)If and , then ;(ii)If and , then ;(iii)If , then . Case 2. If , then and . Case 3. If and , then Hence, . Case 3. If and, then and , hence . Case 3. If and, then and , hence .
Thus, condition () of Definition 5 holds and hence the theorem is proved.
Example 4. Let be an LA-polygroup with the Cayley (Table 4).
Then, is an LA-polygroup with six elements and the Cayley (Table 5), where .
2.2. Extension of an LA-Polygroup by a Set:
Suppose that is an LA-polygroup and , where is a nonempty set such that (ordered from left to right, i.e., ). Put and . We definewhere . The system is known as the extension of LA-polygroup by a set and represented by .
Theorem 2. Let be an LA-polygroup and be a nonempty set such that . Then, is an LA-polygroup.
Proof. Suppose that , , , and .
If , then clearly left invertive law holds.
If , then we consider the following cases: Case 1. If and , then . Case 2. If , , and , then Case 3. If , , in the following way:(i)such that , then(ii)such that , then(iii)such that , thenThus, left invertive law holds. Now, we prove axiom () of Definition 5. Let such that . Then, we consider the following cases: Case 1. If , then either or or :(i)If , then ,(ii)If and , then ; if and , then and ,(iii)If , then . Case 2. If , then and .Thus, condition () of Definition 5 holds and hence the theorem is proved.
Example 5. Let be an LA-polygroup with the Cayley (Table 6).
Then, is an LA-polygroup with five elements and the Cayley (Table 7), where .
This is an LA-polygroup and not a polygroup.
3. Cayley Graphs over LA-Groups and LA-Polygroups
Definition 6. Suppose that is an LA-group and is a subset of such that(i),(ii); then, Cayley graph of relative to is the simple graph which has vertex set and edge set
Example 6. The Cayley graphs of the LA-group given in Example 1, with connection sets and , are shown in Figures 1 and 2. For , we haveFor , we have
Definition 7. Given an LA-polygroup and such that , say the connection set. Then, we define the generalized Cayley graph over LA-polygroup which is the simple graph having vertex set and the edge setIf we have an LA-polygroup and a connection set such that , then the graph is known as a -.
Here, we give few examples of GCLAP-graphs.
Example 7. The generalized Cayley graph of the left almost polygroup , with connection set which is shown in Figure 3, where”” is defined in Table 8.
Example 8. The generalized Cayley graph of the left almost polygroup with connection set which is shown in Figure 4, where”” is defined in Table 9.
4. 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 GCLAP-graph.
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 be an LA-polygroup, where (as defined in Example 2). Then, are isomorphic to the complete graphs on vertices, where . Hence, it is proved.
Lemma 3. Show that each star graph of order at least three is a GCLAP-graph.
Proof. Suppose that , where and is defined in Example 1. Then, Cayley table for is given in Table 10.
Now, by considering connection set , we can see that .
Lemma 4. If is a GCLAP-graph. Then, show that is also a GCLAP-graph, where .
Proof. Let be a GCLAP-graph. Then, we have a left almost polygroup and a connection set such that . Suppose that and . By Extension (II) and Definition 7, we have . Now, by using induction, , where and is a connection set. Hence, . Thus, is a for every .
Definition 8. Let and . Then, the -pseudo complete graph is known as the complement of the graph and represented by . A graph is known as a pseudocomplete graph if for some and .
Example 9. Pseudocomplete graphs on five vertices are shown in Figure 5.
Lemma 5. If and , then the following statements hold:(i) is connected,(ii)If then ,(iii)A connected graph of order is a pseudocomplete graph ; it contains as a subgraph.
Lemma 6. All pseudocomplete graphs are GCLAP-graphs.
Proof. Let be a pseudocomplete graph, where and . Consider the LA-polygroup , where , given in Example 2, and the connection set . Then, is isomorphic to the graph. Hence, .
Definition 9. The expansion of the graph , represented by , is the join of the graph and , i.e., .
Lemma 7. Show that the expansion of a GCLAP-graph is a GCLAP-graph.
Proof. Let , where is a left almost polygroup having elements and is a connection set. Suppose that and . Then, .
Definition 10. Let be a graph, , and . The faulty join graph, represented by , is the graph such that , where and . Moreover, a graph is known as -graph if it is isomorphic to , where such that is a left almost polygroup having elements, ,and is a connection set.
Lemma 8. Every - is a GCLAP-graph.
Proof. Suppose that is a - having vertices. So, we have a left almost polygroup having elements and a connection set such that . Let , then we have
Hence, is a GCLAP-graph.
Up to here, we have determined a few types of GCLAP-graphs. 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 GCLAP-graph) 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 .
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.
Proof. Suppose that is a simple graph of order three, four, and five (except cycle graph of order five). Then, as per the connectivity of , two cases can be thought of: Case 1. If is a connected graph, then we have six subcases: Subcase (i) (cycles and complete graphs): Consider the LA-polygroup as in Example 7. Then, , so is a GCLAP-graph. Also, since , , and are complete graphs, therefore, by Lemma 2, they are GCLAP-graphs. Subcase (ii) (star graphs): Think about LA-polygroup as given in Example 5, then . Also, since and are star graphs, so by Lemma 3, they are GCLAP-graphs. Subcase (iii) (path graphs): Consider the LA-polygroup and LA-group as defined in Examples 7 and 6, respectively. Then, , so is a GCLAP-graph. Also, since , therefore, by Lemma 1, is a . Subcase (iv) (pseudocompleted graphs): Definition 8 implies that , and are pseudocomplete graphs and by Lemma 6, they are . Subcase (v) (expansion graphs): Think about the LA-polygroup as in Example 7 and , given in Table 11. Definition 9 and above cases suggest that the graphs , and are expansion graphs of GCLAP-graphs and by Lemma 7, we infer that they are GCLAP-graphs. Subcase (vi) ( graphs): Consider three LA-polygroups , , and as in Example 7, Example 4, and Example 3, respectively, then the following isomorphisms hold: Thus, the graphs , and are -. By Lemma 8, we conclude that they are GCLAP-graphs. In the end, if we consider the left almost group given in Example 6 and LA-polygroup, with the Cayley (Table 12). Then, by Lemma 1, we conclude that and hence the following isomorphisms hold: Thus, all simple connected 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. Case 2. If is not connected. We have two subcases: Subcase (i): if contains two nontrivial parts, then is isomorphic to , , and . We have , , and , where and are given in Example 3 and Subcase (vi), respectively. Subcase (ii): if is a subgraph of , then , for and is isomorphic to or isomorphic to or a connected graph of order . By Lemma 4, we conclude that is a which completes the proof.
The question that usually comes to mind is what happens if ; we present this query as an open problem [28, 29].
Appendix
There are 28 connected simple graphs of order three, four, and five (except cycle graph of order five) and they are denoted by as given in Figure 6.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors of this paper declare that they have no conflicts of interest.
Acknowledgments
This research was funded by the Deanship of Scientific Research at Princess Nourah Bint Abdulrahman University through the Fast-Track Research Funding Program.