New types of graphs induced by topological spaces

Abstract

In this article, a technique to construct a new type of topological structures by graphs called topological graphs is introduced. We use the concept of a homeomorphism between topological graphs as a topological property to prove the isomorphic between graphs. We construct a computer program that builds graphs and its topological graphs. This model helps in studying many topological properties such as continuity, connectedness, compactness and separation axioms on graphs. Finally, we present some examples of different types of graphs and their topological graphs and study some of their topological and algebraic properties.

Keywords

Graphs a topological graph a topological structure homeomorphic isomorphic

1 Introduction and Preliminaries

The Nobel prize 2016 in physics was gifted to three scientists in phase transitions and topological phases of matter, this event has directed the attention to the need of more knowledge about the topology. Topology is a branch of mathematics whose concepts exist not only in almost all branches of mathematics, but also in many real life applications and concerned with all questions directly or indirectly related to continuity. Many topologists suggested topological models in biology [6 –8] and in medicine [12 , 17].

Topology is a branch of geometry called rubber sheet geometry. It is the science that deals with the properties of things that does not depend on a dimension, which means that it allows increases and decreases, but without cutting things. If X is a nonempty set, a collection τ of subsets of X is said to be a topology on X if the following condition holds X and φ belongs to τ, the finite intersection of any two sets in τ belongs to τ and the union of any number of sets in τ belongs to τ [16].

Graph theory [2 , 18] has recently emerged as a subject in its own right as well as being an important mathematical tool in diverse subjects as operational research, chemistry, sociology and genetics. A graph G = (V, E), is an ordered pair of disjoint sets (V, E), where V is a nonempty set and E is a subset of unordered pairs of V. The vertices and edges of a graph G are the elements V = V (G) and E = E (G), respectively. A graph G is finite (resp. infinite) if the set V (G) is finite (resp. infinite). The degree of a vertex u ∈ V (G) is the number of edges containing u. If there is no edge in a graph G contains a vertex u, then u is called an isolated point, and so the degree of u is zero. An edge which has the same vertex to ends is called a loop, and the edge with distinct end is called a link. A graph is simple if it has no loop and no pair of its links join the same pair of vertices. A graph which has no edge is called a null graph. A simple graph is called complete graph, if any two distinct vertices are joined by an edge. If the vertex set of a graph G can be split into two disjoint sets A and B, so that each edge of a graph G joins a vertex of A and a vertex of B, then a graph G is a bipartite graph. A complete bipartite graph is a bipartite graph in which each vertex in A has been joined to each vertex in B by just one edge. Given a graph G, a walk in G is a finite sequence of edges of the form v₀v₁, v₁v₂, ⋯ , v_n-1v_n. We call v₀ the initial vertex and v_n the final vertex of the walk. A walk in which all the edges are distinct is a trial. If, in addition, the vertices v₀, v₁, v₂, ⋯ , v_n are distinct (except, possibly,(v₀ = v_n), then the trial is a path. A path or trial is closed if v₀ = v_n) and a closed path containing at least one edge is a cycle. We can combine two graphs to make a larger graph. If the two graphs are G₁ = (V (G₁) , E (G₁)) and G₂ = (V (G₂) , E (G₂)), where V (G₁) and V (G₂), are disjoint, then their union G₁ ∪ G₂ is the graph with the vertex set V (G₁) ∪ V (G₂) and edge family E (G₁) ∪ E (G₂). A graph that is in one piece, so that any two vertices are connected by a path is a connected graph, and disconnected otherwise. Clearly, any disconnected graph G can be expressed as the union of connected graphs, each of which is a component of G. In many applications, it is necessary to associate with each edge of a graph an orientation or direction. In some situations, the orientation of the edges is a "true" orientation in the sense that the system represented by the graph exhibits some unilateral property. For example, the direction of the nonway streets of a city and the orientations representing the unilateral property of a communication network are true orientation of the physical system. In other situations, the orientations used a "pseudo"-orientation used in lieu of an elaborate reference system. For example, in electrical network theory the edges of a graph are assigned arbitrary orientations to represent the references of the branch currents and voltages. The study of directed graphs arises from making the roads into one way streets. The connection between graph theory and different subjects such as, in the medicine [15], and in physics [11]. The two sets representing two vertices the graph connected by r edges will intersect in exactly r elements. If v_i and v_j are vertices of a graph G, and A and B are the sets representing v_i and v_j respectively, we denote by N (v_i, v_j) the cardinal number of the set A ∩ B. The aim of the present paper is devoted to establish some new types of graphs in terms of its topological graphs or topological structures. We investigate some methods to generate topological graph from a given a graph and vice versa, not in general. We introduce a new method to prove an isomorphic between graphs through the homeomorphism between its topological graphs. In other words, we study the isomorphic between graphs in the view point of topology. We present some examples of topological graphs for different types of graphs and vice versa. Some topological and algebraic properties will be studied for graphs and topological graphs. We construct an algorithm to calculate the number of edges of graphs through its topological structures. We apply Theorem 3.3 on some special types of graphs and different examples. The adjacency matrix is used to establish our given algorithm for graphs and its topology. It is also used to investigate the degree of each vertex of graphs. In this case, the topological structures of graphs will help to study some properties of graphs such as continuity, connectedness and separation axioms via its topological graphs. The model of topological structures helps also to apply graphs in biology, medicine and many real life problems.

Definition 1.1. [[4, 18]] If G (V, E) is a directed graph and u, v ∈ V, then

(i) u is invertex of v if $\vec{uv} \in E (G)$ .

(ii) u is outvertex of v if $\vec{vu} \in E (G)$ .

(iii) The indegree of a vertex v is the number of vertices u such that $\vec{vu} \in E (G)$ .

(iv) The outdegree of a vertex v is the number of vertices u such that $\vec{vu} \in E (G)$ .

Definition 1.2. [[14]] Let X = {1, 2, 3, ⋯ , n} be a set, a transformation S_n on X can be defined as a bijective function from X to X i.e, f : X → X, the number of transformations of S_n = n !.

Definition 1.3. [[2]] Two graphs G₁ and G₂ are homomorphic if there exists a function F : G₁ → G₂, such that for each v₁, v₂ ∈ V (G₁), and {v₁, v₂} ∈ E (G₁), then {F (v₁) , F (v₂)} ∈ E (G₂).

Definition 1.4. [[2]] Two graphs G and H are isomorphic if there exists a bijective function F : V (G) → V (H), such that for each two adjacent vertices u and v in G, then F (u) and F (v) are also adjacent in H.

Definition 1.5. [[3]] Let X be a topological space. dimX=-1 if X = φ. Let n is a positive integer and dimX ≤ k be defined for each k ≤ n - 1. Then dimX ≤ n if X has a base β such that dim∂B ≤ n - 1, for all B ∈ β, where ∂B is the boundary of a set B.

Definition 1.6. [[12]] A graph G is said to be a topological graph if there exists a topology τ generated by G.

2 The graphic of a class of sets

Definition 2.1. Let C₁ ⊆ P (X), C₂ ⊆ P (Y), C₁ is parallel to C₂ (sayX ∼ Y), if there exists a bijective function F : X → Y such that for each c₁ ∈ C₁, there exist c₂ ∈ C₂ and F (c₁) = c₂. Let X = Y. Then C₁ and C₂ are parallel if there exists a transformation F : X → X, such that F (C₁) = C₂.

Example 2.2. Let X = {1, 2, 3}. Assume that C₁ = {{3} , {1, 2} , {3, 1}}, C₂ = {{1} , {2, 3} , {1, 2}}. Then C₁ and C₂ are parallel. Because of, there exists a bijective function F : X → X, such that F ({3}) = {1}, F ({1, 2}) = {2, 3}, F ({3, 1}) = {1, 2}.

In the following, every class of sets can be represented to a general graph. Any graph G with only one edge can be represented by a class {A, B} such that A ∩ B is a singleton |A ∩ B|=1. If G contains r-edges, then |E (G) | = |A ∩ B| = r, where |A| refers to the cardinality of a set A. So, every graph can be illustrated by different classes as shown in Example 2.3.

Example 2.3. The classes C₁={{1} , {2} , {1, 2}}, C₂={{1, 2, 3} , {4, 5} , {3, 4}}, C₃={{3, 7} , {8, 9, 10} , {3, 10, 12}} of a set X={1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12} represent the same graphs as shown in Fig. 1.

Fig.1

Three classes with only one graph.

Definition 2.4. If C = {C_i : i ∈ I} is the set of all classes of a set X, Then C_i can be represented by the same graph G. The graph number of a graph G equals min {| ⋃ C_i| : i ∈ I}. In other words, if G is a graph, then the graph number of G = m, that means, there is no number r < m such that, | ⋃ _i∈IC_i| = r.

Example 2.5. From Example 2.3, ⋃C₁ = {1, 2}, ⋃C₂ = {1, 2, 3, 4, 5} and ⋃C₃ = {3, 7, 8, 9, 12}. Then | ⋃ C₁|=2, | ⋃ C₂|=5, | ⋃ C₃|=5. This means that the number of graph on C₁ equals 2, and the number of graphs on C₂ and C₃ equals 5.

In the following, we show that every graph G can be illustrated with a class of sets.

Definition 2.6. Let G be a graph and v_i, v_j ∈ V (G). If the vertices v_i is represented by a set A and v_j is represented by a set B in a set graph G, then N (v_i, v_j) is defined by |A ∩ B|. It is clear that N (v_i, U) = |A| and N (v_j, U) = |B|, where U is the universe set of vertices.

Example 2.7. From Example 2.3, if A={1} and B={1, 2}, then N (A, B)=1, for A ∩ B={1} and |A ∩ B|=1.

Theorem 2.8. If G₁ and G₂ are two graphs which represented by two parallel classes C₁ and C₂, respectively, then G₁ and G₂ are isomorphic.

Proof. If C₁ and C₂ are two parallel classes, then there exists a bijective a function F from C₁ to C₂ i.e, F : C₁ → C₂, such that F (C₁) = C₂, where C₁ = {A_i : i ∈ I} and C₂ = {F (A_i) : i ∈ I}. Then every vertex v_{A
_i} in G₁, the graph generated by C₁, have a corresponding vertex v_{F(A_i)} in G₂, the graph generated by C₂. So N (v_{A
_i}, v_{A
_j}) = |A_i ∩ A_j| = |F (A_i ∩ A_j) | = |F (A_i) ∩ F (A_j) | = N (v_{F(A_i)},v_{F(A_j)}).

The converse of Theorem 2.8 may not be true, in general, because every graph can be represented by many classes and may not be parallel. In this case, we use the natural numbers N to represent the class of graphs as shown in Example 2.9.

Example 2.9. In Fig. 2, the two graphs of classes C₁ and C₂ are isomorphic. But the set graphs can not be represented by two parallel classes.

Fig.2

Two isomorphic graphs.

In the following, we describe a new class from any simple graph G.

Definition 2.10. In a graph G, any isolated point can be represented by an empty set φ in a new definite class C.

Definition 2.11. In a graph G, the universal set U can be represented by a subset of natural numbers N. The class of a universal set vertex U is adjacent with each set vertex in its set graph.

Now, we explain that the class of sets in Definition 2.10 and Definition 2.11 represents a graph.

Let A and B are two nonempty sets and |A| = m, |B| = n. The graphs represented by a class {A, B, A ∩ B, A ∪ B} will be given as in the following cases:

Case 1: if A ∩ B = φ, then the graph G is shown in Fig. 3.

Fig.3

Disjoint graph sets.

Case 2: If A ⊂ B, then |A ∩ B| = |A| and |A ∪ B| = |B|. The graph G is shown in Fig. 4.

Fig.4

Inclusion graph sets.

Case 3: If A ∩ B ¬ = φ,|A| ¬ = |B| and |A ∩ B| = r, where r ∈ N, then the graph G is shown in Fig. 5.

Fig.5

Intersection graph sets.

Remark 2.12. Every graph set for a subgraph H of G is a subset from the graph set of G. Each graph set can also be represented by the class P (N), where P (N) denotes to the power set of N.

3 Some operations on topological graphs

In this section, we use a set graph to generate a topological space, we say also, a topological graph or a topological structure. The set graph will be represented by P (N). Some algebraic operations such as, a union, an intersection and a subset, between vertices are defined by:

v_A
₁ ∪ v_A
₂ ∪ v_A
₃ ∪ . . . = v_{A₁∪A₂∪A₃∪...}.

v_A ∩ v_B = v_A∩B.

v_A ⊆ v_B if and only if A ⊆ B.

Definition 3.1. A pseudograph P, with degree n, can be represented by a topology for a set X={1, 2, 3, ⋯ , n}.

Example 3.2. The topological graph on a discrete graphs D will be constructed if we delete only the loops in a pseudograph P. In simple graphs, we draw only one edge between each adjacent vertices in a discrete graph D.

Theorem 3.3.

The number of edges |E_p (G) | of a pseudograph G of the set X = {1, 2, ⋯ , n} equals n2^n-2 (2^n-1 - 1) + n2^n-1.

The number of edges |E_d (G) | of a discrete graph arising from a pseudograph i.e, by deleting loops in P equals n2^n-2 (2^n-1 - 1).

The number of edges |E_s (G) | of a simple graph arising from the discrete graph i.e, by drawing only one edge joint two adjacent vertices in D equals $\frac{1}{2} (2^{2 n} - 2^{n} - 3^{n} + 1)$ .

Proof. The relation between sets can be shown in Fig. 6:

Let A ⊆ X, |A| = m and |A^c| = n - m deg (V_A)=∑R (V_A, V_B)+∑R (V_A, V_E) + ∑R (V_A, V_L)+∑R (V_A, V_K)=∑|A ∩ B|+∑|A ∩ E| +∑| (A ∩ L|+∑|A ∩ K| =∑|B|+∑|A|+∑|A ∩ L|. $\sum | B | = \sum_{i = 1}^{m - 1} i$ ^mC_i= $\sum_{i = 1}^{m - 1} i \frac{m!}{i! (m - i)!}$ = $m \sum_{i = 1}^{m - 1} \frac{m - 1)!}{(i - 1)! (m - i)!}$ = ! $m \sum_{i = 1}^{m - 1}$ ^m - 1C_i-1 = $m \sum_{j = 0}^{m - 2} ($ ^m - 1C_j - 1) = m (2^m-1 - 1) , since B ⊆ A, then∑|A| = m (2^m-1 - 1). But A ⊆ E, then by complement ∑|A|=m (2^n-m - 1) and ∑|A ∩ L|=(2^n-m - 1) $\sum_{i = 1}^{m - 1} i$ ^mC_i =m (2^n-m - 1) (2^m-1 - 1). So deg (V_A) = m (2^n-1 - 1) +2m. It follows that the number of edges of pseudographs G equals |E_p (G) | = $\frac{1}{2} \sum_{m = 1}^{n}$ ⁿC_mdeg (V_A) = $\frac{1}{2} \sum_{m = 1}^{n}$ ⁿC_m (m (2^n-1 - 1)+2m) = $\frac{1}{2} \sum_{m = 1}^{n}$ ⁿC_mm (2^n-1 - 1) + $\frac{1}{2} \sum_{m = 1}^{n}$ ⁿC_m2m =n2^n-2 (2^n-1 - 1)+ $\sum_{m = 1}^{n} m$ ⁿC_m =n2^n-2 (2^n-1 - 1) + $n \sum_{m = 1}^{n}$ ^n - 1C_m-1 = n2^n-2 (2^n-1 - 1)+ $n \sum_{j = 0}^{n - 1}$ ^n - 1C_j = n2^n-2 (2^n-1 - 1)+n2^n-1.

deg (V_A) = m (2^n-1 - 1). It follows that the number of edges of discrete graphs equals |E_d (G) | = $\frac{1}{2} \sum_{A} \deg (V_{A})$ = $\frac{1}{2} \sum_{A}$ ⁿC_mm * (2^n-1 - 1) = $\frac{1}{2} (2^{n - 1} - 1) \sum_{m = 1}^{n} m$ ⁿC_m = $\frac{1}{2} (2^{n - 1} - 1) \sum_{m = 1}^{n} m$ ⁿC_m = $\frac{1}{2} (2^{n - 1} - 1) \sum_{m = 1}^{n} m \frac{n!}{m! (n - m)!}$ = $\frac{n}{2} (2^{n - 1} - 1) \sum_{m = 1}^{n} \frac{(n - 1)!}{(m - 1)! (n - m)!)}$ = $\frac{n}{2} (2^{n - 1} - 1) \sum_{m = 1}^{n}$ ^n - 1C_m-1 = $\frac{n}{2} (2^{n - 1} - 1) \sum_{j = 0}^{n - 1}$ ^n - 1C_j = $\frac{n}{2} (2^{n - 1} - 1) 2^{n - 1}$ =n2^n-2 (2^n-1 - 1).

deg (V_A) =2ⁿ - 2^n-m - 1. It follows that the number of edges of simple graphs G equals |E_s (G) | = $\frac{n}{2} \sum_{m = 1}^{n}$ ⁿC_mdeg (V_A) = $\frac{1}{2} \sum_{m = 1}^{n}$ ⁿC_m (2ⁿ - 2^n-m - 1) = $\frac{1}{2} (2^{n} - 1) \sum_{m = 1}^{n} n C_{m} - \frac{1}{2} 2^{n} \sum_{m = 1}^{n}$ ⁿC_m2^-m = $\frac{1}{2} (2^{n} - 1) (2^{n - 1} - 1) - 2^{n - 1} \sum_{m = 1}^{n} (\frac{1}{2})^{m}$ ⁿC_m = $\frac{1}{2} (2^{2 n} - 2^{n + 1} + 1) - 2^{n - 1} ((1 + \frac{1}{2})^{n} - 1)$ = $\frac{1}{2} ((2^{2 n} - 2 * 2^{n} + 1) - (3^{n} - 2^{n}))$ = $\frac{1}{2} (2^{2 n} - 2^{n} - 3^{n} + 1)$ .

□

We apply Theorem 3.3 in Example 3.4.

Fig.6

Relationship Between Sets.

Example 3.4. The number of edges of pseudographs G, discrete graphs and simple graphs of a set X={1, 2, 3} is given by:

|E_P (G) |=3 * 2^3-2 (2^3-1 - 1) +3 * 2^3-1 =3 * 2 *3 + 16=18 + 12=30.

|E_D (G) |=3 * 2^3-2 (2^3-1 - 1)=3 * 2 *3=18.

|E_S (G) |= $\frac{1}{2} (2^{6} - 2^{3} - 3^{3} + 1)$ =15, respectively, where S denotes to simple graphs.

Theorem 3.5. Let G be a graph satisfies the following conditions:

G contains only one isolated vertex v₀.

G contains a vertex v adjacent with each vertex in G ∖ {v₀} and N (v_i, v) ≤ N (v_i, U) ≤ N (v, U), for every v_i ∈ G ∖ {v₀}.

For every two distinct vertices v_i,v_j ∈ V (G), then v_i ∪ v_j ∈ V (G) and v_i ∩ v_j ∈ V (G). Then the class τ of vertices is a topological graph.

Proof. Let G be a graph. To complete the proof, it is sufficient to prove three conditions of topologyon τ.

By Definition 2.4, G can be represented by different classes. Suppose that the graph number of G is m. Then there exists a class, say τ, such that τ ⊆ P ({1, 2, 3, ⋯ , m}) =2^{{1,2,3,⋯,m}} and represents G. Each A_i ∈ τ represents v_i and X ∈ τ represents the set of vertices V (G). So, by Definition 2.6, N (v_i, V (G)) = N (v_i, U). If N = {1, 2, 3, . . . , m}, then |A ∩ X| = |A ∩ N| = |A|. This means that each A ∈ τ satisfies that A ⊆ X. Since every vertex in G is a graph subset of V (G), i.e, for every v_i ∈ V (G) the singleton {v_i} ⊆ V (G) and each element in τ is a subset from X ∈ τ, then X = {1, 2, . . . , m}. Also, the isolated vertex v₀ can be represented by φ ∈ τ.

Let v₁, v₂, ⋯ be an arbitrary different vertices in V (G) represented by A_v
₁, A_v
₂, ⋯. Since v₁∪ v₂ ∪ ⋯= v₍A_v
₁ ∪ A_v
₂ ∪ ⋯), then A_v
₁ ∪ A_v
₂ ∪ ⋯ ∈ τ.

If v_i and v_j are two different vertices in V (G) and represented by A_i and A_j, respectively. Since v_i ∩ v_j=v_{A_i∩A_j}, then A_i ∩ A_j ∈ τ. Therefore τ is a topology.□

Now, we use the topological graph τ to calculate the number of edges in G using Theorem 3.6.

Theorem 3.6. The number of edges of a topological graph represented by a topology τ={φ, {1} , {1, 2} , ⋯ , {1, 2, ⋯ , n}} is given by: $\frac{1}{6} n (n^{2} - 1)$ .

Proof. Let A_m

={1, 2, 3, ⋯ , m}, 1 < m < n, degV_{A
_m}

=m (n - m)+ $\sum_{r = 1}^{m - 1} r$

= m (n - m)+ $\frac{1}{2} m (m - 1)$

=mn - m²+ $\frac{1}{2} m^{2} - \frac{1}{2} m$

= $mn - \frac{1}{2} m^{2} - \frac{1}{2} m$ .

The number of edges is

|Edges|= $\frac{1}{2} \sum_{m = 1}^{n} \deg V_{A_{m}}$

= $\frac{1}{2} \sum_{m = 1}^{n} (mn - \frac{1}{2} m^{2} - \frac{1}{2} m)$

= $\frac{1}{2} n \sum_{m = 1}^{n} m - \frac{1}{4} \sum_{m = 1}^{n} m^{2} - \frac{1}{4} \sum_{m = 1}^{n} m$

= $n^{2} (n + 1) - \frac{1}{24} n (n + 1) (2 n + 1)$ $- \frac{1}{8} n (n + 1)$

= $\frac{1}{24} n (n + 1) (6 n - 3 - 2 n - 1)$ = $\frac{1}{24} n (n + 1) (4 n - 4)$ = $\frac{1}{6} n (n^{2} - 1)$ =^n +1C₃ .

□

To explain the results of Theorems 3.5, 3.6, we present Examples 3.7, 3.8.

Example 3.7. Let X = {1, 2, 3} with a topological graph τ = {φ, {1} , {1, 2} , X}. The number of edges equals 4. See Fig. 7.

Fig.7

A topological graph for 3 points.

Example 3.8. Let Y = {1, 2, 3, 4} with a topological graph τ={φ, {1} , {1, 2} , {1, 2, 3} , Y}. The number of edges equals 10. See Fig. 8.

Fig.8

A topological graph for 4 points.

Remark 3.9. Every topological graph can be illustrated by a graph, but the converse may not true, in general, as in Examples 3.10, 3.11.

Example 3.10. Let G be a graph and its topological graph τ be shown in Fig. 9. We construct τ by the following procedures: An isolated vertex v_f represented by φ; the vertex v_a and v_b or one of them can be considered as a vertex adjacent with all vertices except v_f. Take v_a is a vertex V represented by a set X. Since N (v_e, V (G)) =1, then v_e represented by {1}. At N (v_d, V (G)) =2, then v_d represented by {1, 2}. Also, at N (v_c, V (G)) =2, then v_c represented by {1, 3}. In the same manner, v_b represented by {1, 2, 3} and v_a represented by {1, 2, 3, 4}, for N (v_b, V (G)) < N (v, U), for all v in G. Therefore τ={φ, {1} , {1, 2} , {1, 3} , {1, 2, 3} , X} is a topology on X. τ is said to be a topological graph of G.

Fig.9

A graph which is a topological graph.

Fig.10

A graph which is not a topological graph.

Example 3.11. The structure in Fig. 11 is not a topological graph of any graph G, this means that, the class τ of G does not satisfies conditions of topology. Because of deg (v_a)= 4 (where deg denotes to the degree of vertex), deg (v_b) =3 = deg (v_c). Now, v_a represented by X = {1, 2, 3, 4}, v_b represented by a set, say {2}, and v_c represented by a set, say {3}. Although N (v_b, v_c) =1, we have {2} ∩ {3}=φ. Therefore τ= {φ, X, {2} , {3}} is not a topology, for {2} ∪ {3}= {2, 3} ∉ τ.

4 A topological graph algorithm

In this section, we give an algorithm which describes graphs and its topological graphs. We use the topological dimension [3] as a tool for this algorithm. In this way, we calculate the degree of dimension which used to get the number of edges for graphs. In the algorithm below, d denotes to dimension, n = |X| and m = |A|, for any A ⊆ X.

Algorithm:

Input: d, n and m.

Output: Degree of each vertex.

1. Insert n,m.

2. for i ∈ n

A_i =insert d_i.

end

3. for i ∈ n

topology + = A_i.

end

4. Function(For a discrete graph) degV_A∩B = m (2^m-1 - 1).

degVA ∩ E = m (2^n-m - 1).

degV_A∩L = m (2^n-m - 1) (2^m-1 - 1).

degV_A = m (2^n-1 - 1).

|Edges| = n2^n-2 (2^n-1 - 1).

end

5. Function(For a pseudograph) degV_A = m (2^n-1 - 1) +2m.

|Edges| = n2^n-2 (2^n-1 - 1) + n2^n-1.

end

6. Function(For a simple graph)

degV_A = 2ⁿ - 2^n-m - 1.

$| Edges | = \frac{1}{2} (2^{2 n} - 2^{n} - 3^{n} + 1)$ .

end

7. for i ∈ 2ⁿ

degV_i = i * n.

end

5 An isomorphic between topological graphs

In this section, we study the isomorphic between graphs, which is an important property in graph theory, through the homeomorphic between its topological graphs which is a topological property. In our study, we use the topological graph stated in Section 3. The following examples discuss special types of graphs such as a Peterson graph in Fig. 11 and some other graphs in Fig. 12.

Fig.11

Two isomorphic graphs G₁ and G₂.

Fig.12

Two isomorphic graphs G₁ and G₂.

Example 5.1. The isomorphism by topological graphs is shown in Fig. 11. From Table 1, we have a class C = {{1, 2, 3} , {1, 4, 5} , {4, 6, 7} , {6, 8, 9} , {3, 9, 10} , {2, 12, 13} , {5, 14, 15} , {7, 11, 12} , {8, 13, 14} , {10, 11, 15}}, which represents a graph G₁. By the same manner, we have the same class by Table 2, which represents a graph G₂. Table 3 proves the homeomorphic between topological graphs of G₁ and G₂.

Table 1

A topological graph of G₁

Step/Vertex	1	2	3	4	5	6	7	8
u ₁	{1, 2, 3}
u ₂	{1,	{1, 4, 5}
u ₃		{4,	{4, 6, 7}
u ₄			{6,	{6, 8, 9}
u ₅	{3,			{3, 9,	{3, 9, 10}
u ₆	{2,					{2, 12, 13}
u ₇		{5,					{4, 14, 15}
u ₈			{7,			{7, 12,		{7, 11, 12}
u ₉				{8,		{8, 13,	{8, 13, 14}
u ₁₀					{10,		{10, 15,	{10, 11, 15}

Table 2

A topological graph of G₂

Step/Vertex	1	2	3	4	5	6	7	8
v ₁	{1, 2, 3}
v ₂	{1,	{1, 4, 5}
v ₃		{4,	{4, 6, 7}
v ₄			{7,	{7, 11, 12}
v ₅				{11,	{10, 11, 15}
v ₆	{3,				{3, 10	{3, 9, 10}
v ₇			{6,			{6, 9,	{6, 8, 9}
v ₈							{8,	{8, 13, 14}
v ₉	{2,				{2, 12,			{2, 12, 13}
v ₁₀		{5,			{5, 15			{5, 14, 15}

Table 3

Homeomorphic between topological graphs of G₁ and G₂

Set	G ₁	G ₂
{1, 2, 3}	u ₁	v ₁
{1, 4, 5}	u ₂	v ₂
{4, 6, 7}	u ₃	v ₃
{6, 8, 9}	u ₄	v ₇
{3, 9, 10}	u ₅	v ₆
{2, 12, 13}	u ₆	v ₉
{5, 14, 15}	u ₇	v ₁₀
{7, 11, 12}	u ₈	v ₄
{8, 13, 14}	u ₉	v ₈
{10, 11, 15}	u ₁₀	v ₅

Since there are a homeomorphic between topological graphs in Table 3, then G₁ and G₂ are isomorphic in Fig. 11.

Example 5.2. Let G₁ and G₂ be two graphs which is shown in Fig. 12. From Tables 4, 5, we have the same topological graph τ={φ, {1} , {2} , {3} , {1, 2} , {1, 3} , {2, 3} , {1, 2, 3}}, for graphs G₁ and G₂. We proves that G₁ is an isomorphic with G₂, inTable 6.

Table 4

A topological graph of G₁

Step/Vertex	1	2	3	4
u ₀	φ
u ₁		{1}
u ₂			{2}
u ₃				{3}
u ₄		{1,	{1, 2}
u ₅		{1,		{1, 3}
u ₆			{2,	{2, 3}
u ₇		{1,	{1, 2,	{1, 2, 3}

Table 5

A topological graph of G₂

Step/Vertex	1	2	3	4
v ₀	φ
v ₁		{1}
v ₂			{2}
v ₃				{3}
v ₄			{2,	{2, 3}
v ₅		{1,		{1, 3}
v ₆		{1,	{1, 2}
v ₇		{1,	{1, 2,	{1, 2, 3}

Table 6

Homeomorphic between two topological graphs of G₁ and G₂

Sets	G ₁	G ₂
φ	u ₀	v ₀
{1}	u ₁	v ₁
{2}	u ₂	v ₂
{3}	u ₃	v ₃
{1, 2}	u ₄	v ₆
{1, 3}	u ₅	v ₅
{2, 3}	u ₆	v ₄
{1, 2, 3}	u ₇	v ₂

Since there are a homeomorphic between topological graphs in Table 6, then G₁ and G₂ are isomorphic in Fig. 12.

6 Conclusion and discussion

The field of mathematical science which goes under the name of the topology is concerned with all questions directly or indirectly related to topological graphs. Therefore, the theory of graphs and topological spaces became the most important subjects in mathematics. On the other hand, topology plays a significant rule in quantum physics, high energy physics and superstring theory [5]. Thus we study the topological structure of graphs and calculate the degree of vertices and the number of edges of graphs which may have possible applications in quantum physics and superstring theory. Moreover, the concepts proposed in this paper can be extended in fuzzy topological structures [11] and thus one can get a more affirmative solution in decision making problems [9 , 19–21] in real life solutions.

Footnotes

Acknowledgments

The authors would like to thank the Editors and Reviewers for careful and thorough reading of this manuscript and for the thoughtful comments and constructive suggestions, which help to improve the quality of this manuscript.

References

Akram

, m-Polar Fuzzy Graphs, Studies in Fuzziness and Soft Computing, Springer, 2019.

Bondy A.J., Murty R.S.U., Graph theory with applications, Elsevier Science Publishing Company Inc, 1975.

Cain L.G., Introduction to general, Addison Wesley Publishing Company, Inc, 1994.

Diestel

, Graph theory II, Heidelberg, Springer-Verlag IV, 2010.

El-Naschie

S.M.

, Topics in the mathematical physics of Einfinity theory, Chaos, Solitons, Fractals 30 (2006), 656–663.

El-Sharkasy

M.M.

, Badr

S.M.

, Topological spaces via phenotype- genotype spaces, International Journal of Biomathematics 9(4) (2016), 1–16.

El-Sharkasy

M.M.

, Badr

S.M.

, Modeling DNA and RNA mutation using mset and topology, International Journal of Biomathematics 11(04) (2018).

El-Sharkasy

M.M.

, Shokry

, Separation axioms under crossover operator and its generalized, International Journal of Biomathematics 9(4) (2016), 1–23.

Jiang

, Zhan

, Chen

, Covering based variable precision (I, T)-fuzzy rough sets with applications to multi-attribute decision-making, IEEE Transactions on Fuzzy Systems (2018) doi:10.1109/TFUZZ.2018.2883023.

10.

Chartrand G., Lesniak L., Zhang P., Textbooks in Mathematics(Graphs and Digraphs), Sixth Edition, 2016.

11.

Galton

, A generalized topological view of motion in discrete space, Theoretical Computer Science 305 (2003), 111–134.

12.

Nada

S.I.

, El-Atik

A.A.

, Atef

, New types of topological structures via graphs, Mathematical Methods in the Applied Sciences 41 (2018), 5801–5810.

13.

Nawar

A.S.

, El Atik

A.A.

, A model of a human heart via graph nano topological spaces, International Journal of Biomathematics 12(1) (2019).

14.

Ramji

, Algebra 1 Groups, Rings, Fields and Arithmetic, Springer Nature Singapore Pte Ltd, 2017.

15.

Stadler

B.M.R.

, Stadler

P.F.

, Generalized topological spaces in evolutionary theory and combinatorial chemistry, Journal of Chemical Information and Computer Sciences 42 (2002), 577–585.

16.

Morris S.A., Topology without tears, version of March 1, 2017.

17.

Shokry

, Aly

R.E.

, Topological properties on graph VS medical application in human heart, International Journal of Applied Mathematics 15 (2013), 1103–1109.

18.

Wilson R.J., Rough sets, Introduction to graph theory, Fourth Edition, Longmon Maleysia, 1996.

19.

Zhan

, Sun

, Alcantud

J.C.R.

, Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making, Information Sciences 476 (2019), 290–318.

20.

Zhang

, Zhan

, Wu

, Alcantud

J.C.R.

, Fuzzy β-covering based (I; T)-fuzzy rough set models and applications to multi-attribute decision-making, Computers & Industrial Engineering 128 (2019), 605–621.

21.

Zhang

, Zhan

, Xu

, Covering-based generalized IF rough sets with applications to multi-attribute decision-making, Information Sciences 478 (2019), 275–302.