Constructing New Topological Graph with Several Properties

In this paper, a new idea to configure a special graph from the discrete topological space is given. Several properties and bounds of this topological graph are introduced. Such that if the order of the non-empty set equals two, then the topological graph is isomorphic to the complete graph. If the order equals three, then the topological graph is isomorphic to the complement of the cycle graph. Our topological graph has 𝑛 − 1 complete induced subgraphs with order 𝑛 or more. It also has a cycle subgraph. In addition, the clique number is obtained. The topological graph is proved simple, undirected, connected graph. It has no pendant vertex, no isolated vertex and no cut vertex. The minimum and maximum degrees are evaluated. So , the radius and diameter are studied here


Introduction
A graph is denoted by = ( , ) such that ( ) is a set of all vertices, and ( ) is a set of all edges in . The maximum degree for any vertex of ( ) is denoted by ∆( ) and the minimum degree is denoted by ( ). For any two vertices , are adjacent if there is an edge between them. A complete graph that has order such that each vertex in it is adjacent to − 1 of the remaining vertices. Graph is called a connected graph if for any two vertices ISSN: 0067-2904

Jwair and Abdlhusein
Iraqi Journal of Science, 2023, Vol. 64, No. 6, pp: 2991-29992992 belonging to it there is a path between them. The induced subgraph of , denoted by [ ], is constructed by all vertices of ⊆ ( ) and all edges inside it. For more information, see [1-12, 13, 14, 15]. A clique is a complete induced subgraph of . Clique number ( ) is the number of all vertices for a minimum complete induced subgraph [16]. A discrete topological space is denoted by ( , ), such that is a non-empty set and is a family of all subsets of . There are many papers that studied the transformation of graphs into a topology. There are few studies that convert the topology into a graph, although it appears with different graphics and different properties and many results with more information can be found about these topics in [17][18][19][20]. In our study which is different from the previous studies, we develop a new definition that includes converting a type of topology, which is the discrete topology into a graph. This graph is denoted by = ( , ), where ( ) is the set of all vertices in that includes all subsets of unless , ∅. And ( ) is a set of all edges between any two vertices which are not subset with each other. Several results are also proved, namely, the graph contains complete induced subgraphs. In addition, the minimum degree and the maximum degree of a graph are calculation. The clique number is calculated, and the value of diameter and radius is found.

Definition and Properties of Topological graph
In this section, a new method is applied to the discrete topological space to construct a topological graph. Several important properties of the topological graph are introduced and proved.      Proof: Let be a set of all vertices of singleton elements such that | | = , let , ∈ . Since is not a subset of and is not a subset of for all elements of , then is adjacent to .
Hence, [ ] is a complete induced subgraph of order , so that [ ] = . Let ˊ be a set of all vertices that have two elements such that |ˊ| = ( 2 ), let 1 , 2 ∈ˊ. Since 1 is not a subset of 2 and 2 is not a subset of 1 for all elements of ˊ. Then, 1 is adjacent to 2 . Therefore, [ˊ] is a complete induced subgraph of order ( 2 ), thus [ˊ] = ( 2 ) and so on. Also, if ˊˊ is a set of all vertices that have − 1 elements and for any 1 , 2 ∈ˊˊ. Since 1 is not a subset of 2 and 2 is not a subset of 1 for all elements of ˊˊ, then 1 is adjacent to 2 . Hence, [ˊˊ] is a complete induced subgraph of order ( − 1 ) = , thus [ˊˊ] = ( − 1 ) = . Now, if , any two vertices in and is not adjacent to , then either ⊆ or ⊆ . So, there is 2 null graph between them. Thus, there is no complete induced subgraph between them. Therefore, the graph has − 1 complete induced subgraphs as an example, see Figure 3. 2) The clique number of a graph is ( ) = ( ⌈ 2 ⌉ ) = ( ⌊ 2 ⌋ ).
Proof: 1) If = 2, then by Proposition 2.2, ≡ 2 . It is clear that the degree of every vertex is one. Now, if ≥ 3, let be any vertex in . Then, from proof of Proposition 2.3, and Proposition 2.7, the vertex is adjacent to more than two vertices. So that, ( ) > 2, and has no pendant vertex. 2) Let be a set of all vertices that have ⌈ 2 ⌉ elements (also it is equal to a set have all vertices of ⌊ 2 ⌋ elements), such that | | = ( ⌈ 2 ⌉ ). The set constructs an induced subgraph of order ( ⌈ 2 ⌉ ) isomorphic to the complete graph .

Lemma 2.9 [21]:
If is a graph in which the degree of each vertex is at least two, then has a cycle.
Theorem 2.11: Let be a discrete topological graph of a non-empty set , then is a connected graph.
Proof: From Proposition 2.7, the vertex set is partitioned into − 1 subsets, each subset constituting a complete subgraph. Let and be any two different vertices, then there are two cases one of them if these vertices belong to the same set, so these vertices are connected. In the second case when the two vertices belong to different sets, the first subset is the set of singleton elements and for each element in other vertices, there is at least one element in that is adjacent to it. Thus, the graph is connected.

Proposition 2.12:
Let be a discrete topological graph, then there is no isolated vertex in a graph .
Proof: Since be a connected graph by Theorem 2.11, then the graph has no isolated vertex. Proposition 2.13: Let be a discrete topological graph of a non-empty set , then is a simple graph.
Proof: Assume that and are any two vertices in since ⊆ for all elements of , then there is no edge that joins a vertex to itself. Hence, there is no loop in a graph . If ⊈ and ⊈ , by Definition 2.1. There is an edge between and (only one edge). Thus, there is no multiple edges between them. Proposition 2.14: Let be a discrete topological graph of a non-empty set , then is an undirected graph.
Proof: By definition of a discrete topological graph, the proof is obtained.

3) ( ) = ∆( ) = 3, if = 3.
Proof: 1) If = 2, then according to Proposition 2.2, ≡ 2 so that any vertex in say , then ( ) = 1. Now, if ≥ 3 since the graph has no isolated vertex by Proposition 2.12, and has no pendant vertex by Proposition 2.8. So that, for each vertex in say 1 , then ( 1 ) ≥ 2. A minimum degree of the graph exists to each vertex of a singleton element (and it is equal to the degree of any vertex having − 1 elements). To find a degree of the vertex of a singleton element say , since is adjacent to every vertex of a singleton element from proof of Proposition 2.7. Thus, the number of these vertices is ( − 1 1 ), so ( ) ≥ − 1. The vertex is adjacent to all vertices that have two elements which are not subset of and is not subset of them, and the number of these vertices is ( − 1 2 ). Again, the vertex is adjacent to all vertices that have three elements which are not subset of and is not subset of them, and the number of these vertices is ( − 1 3 ) and so on. The last step when the vertex is adjacent to the vertices that have − 1 elements where the number of these vertices is ( − 1 − 1 ) = 1. Therefore, ( ) = ∑ ( − 1 ) −1 =1 . As an example, see Fig. 2

.3.
To prove a minimum degree in each vertex that have singleton element (equal to degree of any vertex that have − 1 elements). Since all subsets of ( ) of order ( ) form a complete induced subgraph isomorphic to ( ) for = 1, 2 3, … , − 1, by proof of Proposition 2.7.
2) In the graph the maximum degree is founded in each vertex that have ⌈ 2 ⌉ elements (also it is found in each vertex have ⌊ 2 ⌋ elements where the result is equal in both cases). We prove a maximum degree in the vertex of ⌈ 2 ⌉ elements. Since all vertices that have ⌈ 2 ⌉ elements form an induced subgraph of order ( ⌈ 2 ⌉ ) isomorphic to the complete graph . Therefore, . As an example, see Proof: Let be a set of all vertices that have a singleton element, and let any vertex in a graph . Since there is no vertex in is adjacent to all vertices in , then ( ) ≥ 2, so there are two cases as follows: Case 1: If ∈ and any vertex in a graph is not adjacent to , thus there is at least one vertex in say 1 is adjacent to . Since 1 is adjacent to from proof of Proposition 2.7, then Case 2: If ∉ , then again there is a vertex say not adjacent to it, so there are two subcases as follows: Subcase 1: If ∈ , then ( , ) = 2 . This is similar to Case 1, thus ( ) = 2. Subcase 2: If ∉ , then there is at least one vertex in adjacent to and . Hence, ( , ) = 2. From above, ( ) = 2 in each cases. Therefore, ( ) = ( ) = 2.
Proposition 2.18: Let be a discrete topological graph, then has no cut vertex.
Proof: Let any vertex in a graph , if is a vertex of singleton element. Since adjacent with each vertex of singleton element from proof of Proposition 2.7, and form but when the vertex is removed from a graph . Every path between any two vertices 1 , 2 ∈ pass through can be pass through another vertex of singleton element. Then, the graph − is a connected graph, so that is not cut vertex. Again, if is a vertex that has two elements, since is adjacent to each vertex that has two elements from proof of Proposition 2.7, and form ( 2 ) . Then, when the vertex is removed from a graph . Every path between any two vertices 3 , 4 ∈ pass through can be pass through another vertex that have two elements. Then, the graph − is a connected graph, thus is not cut vertex, and so on. And if is a vertex that has − 1 elements, since the set of all vertices that have − 1 elements make an induced subgraph isomorphic to ( −1 ) = from proof of Proposition 2.7. Hence, − is a connected graph and is not cut vertex, and has no a cut vertex.

Conclusions
The aim of this paper is to construct the topological graph from discrete topology. And study many properties of this graph such as it is a simple graph, undirected graph and connected graph. Also, it has no pendant vertex, no isolated vertex and no cut vertex. Also, the minimum degree and the maximum degree are founded. Further, the diameter and the radius of are proved. Finally, the clique number of it is studied.

Open problems
Study more properties of the topological graph, and apply many parameters domination on it. Like: independent domination, total domination, weak domination, strong domination, coindependent domination and bi-domination.

Acknowledgements
We would thank and appreciate the authors of references that we have used in this paper.