Folding on the Chaotic Graph Operations and Their Fundamental Group

*Mohammed Abu Saleem*

### **Abstract**

Our aim in the present chapter is to introduce a new type of operations on the chaotic graph, namely, chaotic connected edge graphs under the identification topology. The concept of chaotic foldings on the chaotic edge graph will be discussed from the viewpoint of algebra and geometry. The relation between the chaotic homeomorphisms and chaotic foldings on the chaotic connected edge graphs and their fundamental group is deduced. The fundamental group of the limit chaotic chain of foldings on chaotic. Many types of chaotic foldings are achieved. Theorems governing these relations are achieved. We also discuss some applications in chemistry and biology.

**Keywords:** chaotic graph, edge graph, chaotic folding, limit folding fundamental group

**2010 Mathematics Subject Classification:** 51H20, 57N10, 57M05, 14F35, 20F34

### **1. Introduction and definitions**

During the past few decades, examinations of social, biological, and communication networks have taken on enhanced attention throughout these examinations; graphical representations of those networks and systems have been evident to be terribly helpful. Such representations are accustomed to confirm or demonstrate the interconnections or relationships between parts of those networks [1, 2].

A graph is an ordered G = (V(G), E(G)) where V(G) 6¼ φ, E(G) is a set disjoint from V(G), elements of V(G) are called the vertices of G, and elements of E(G) are called the edges. The foundation stone of graph theory was laid by Euler in 1736 by solving a puzzle called Königsberg seven-bridge problem as in **Figure 1** [1, 3].

There are many graphs with which one can construct a new graph from a given graph or set of graphs, such as the Cartesian product and the line graph. A graph G is a finite non-empty set V of objects called vertices (the singular is vertex) together with a set E of two-element subsets of V called edges. The number of vertices in a graph G is the order of G, and the number of edges is the size of G. To indicate that a graph G has vertex set V and edge set E, we sometimes write G = (V, E). To emphasize that V is the vertex set of a graph G, we often write V as V(G). For the same reason, we also write E as E(G). A graph H is said to be a subgraph of a graph G if V(H) ⊆ V(G) and E(H) ⊆ E(G). The complete graph with n-vertices will be denoted by *Kn:* A null graph is a graph containing no edges; the null graph with

**2. The main results**

First, we will introduce the following:

*DOI: http://dx.doi.org/10.5772/intechopen.88553*

*edges. The number of chaotic edges is the size of G*.

 ⊆*E G* .

*edge* ½ � *u; v of G,* ½ � *φ*ð Þ *u ; φ*ð Þ*v is a chaotic edge of H*.

∗ *π*<sup>1</sup> G2

∗ *π*<sup>1</sup> G2

.

*G*<sup>2</sup> are of same chaotic homotopy type, it follows that

*chaotic isomorphism) as in* **Figure 3**.

*and E H*

*every chaotic vertex x of H.*

*chaotic proper subgraph*.

<sup>¼</sup> *<sup>π</sup>*<sup>1</sup> G1

≈*π*<sup>1</sup> G1

*π*<sup>1</sup> G1 ⊻ G2

*π*<sup>1</sup> G1 ⊻ G2

**Figure 2.** *Chaotic edge.*

**53**

*V H*

⊆*V G*

**Case 1** (1) *e*1*, e*2*, e*3*,* … *are of the same physical properties*.

*Folding on the Chaotic Graph Operations and Their Fundamental Group*

*sents density, e*<sup>2</sup> *represents hardness, e*<sup>3</sup> *represents magnetic fields, and so on*.

**Definition 5.** *Let G and H be two chaotic graphs. A function φ* : *V G*

**Theorem 1.** *Let G*<sup>1</sup> *and G*<sup>2</sup> *be two chaotic connected graphs. Then*

*:* Hence, *π*<sup>1</sup> G1 ⊻ G2

**Definition 1.** *The chaotic edge e is a geometric edge e*<sup>1</sup> *that carries many other edges* ð Þ *e*2*;e*3*;* … *, each one of them homotopic to the original one as in* **Figure 2***. Also the chaotic vertices of e are v* ¼ ð Þ *v*1*; v*2*;* … *and u* ¼ ð Þ *u*1*; u*2*;* … *. For chaotic edge e, we have two cases*:

**Case 2** (2) *e*1*, e*2*, e*3*,* … *represent different physical properties; for example, e*<sup>1</sup> *repre-*

**Definition 2.** *A chaotic graph G is a collection of finite non-empty set V of objects called chaotic vertices together with a set E of two-element subsets of V called chaotic*

**Definition 3.** *Given chaotic connected graphs G*<sup>1</sup> *and G*<sup>2</sup> *with given edges e*<sup>1</sup> ∈ *G*<sup>1</sup> *and e*<sup>2</sup> ∈ *G*2*, then the chaotic connected edge graph G*<sup>1</sup> *G*<sup>2</sup> *is the quotient of disjoint union G*<sup>1</sup> ∪ *G*<sup>2</sup> *acquired by identifying two chaotic edges e*<sup>1</sup> *and e*<sup>2</sup> *to a single chaotic edge (up to*

**Definition 4.** *A chaotic graph H is called a chaotic subgraph of a chaotic graph G if*

*chaotic homomorphism from G to H if it preserves chaotic edges, that is, if for any chaotic*

**Definition 7.** *A chaotic core is a chaotic graph which does not chaotic retract to*

**Proof.** Let *G*<sup>1</sup> and *G*<sup>2</sup> be two chaotic connected graphs. Since *G*<sup>1</sup> ⊻ *G*<sup>2</sup> and *G*<sup>1</sup> ∨

<sup>¼</sup> *<sup>π</sup>*<sup>1</sup> G1

∗ *π*<sup>1</sup> G2

.

**Definition 6.** *A chaotic folding of a graph G is a chaotic subgraph H of G such that there exists a chaotic homomorphism f* : *G* ! *H, called chaotic folding with f*ð Þ¼ *x x for*

! *<sup>V</sup> <sup>H</sup>*

*is*

n-vertices is denoted by *Nn:* A cycle graph is a graph consisting of a single cycle, the cycle graph with n-vertices is denoted by *Cn:* The path graph is a graph consisting of a single path; the path graph with n-vertices is denoted by Pn [1–11]. Let G and H be two graphs. A function φ : V Gð Þ! V Hð Þ is a homomorphism from G to H if it preserves edges, that is, if for every edge *e*∈*E G*ð Þ*,f e*ð Þ∈*E H*ð Þ [12, 13]. A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a unique core [7].

The folding is a continuous function *f* : *G* ! *H* such that for each *v*∈*V G*ð Þ*, f*ð Þ v ∈*V H*ð Þ*,* and for each *e*∈*E G*ð Þ*,f e*ð Þ∈*E H*ð Þ [14]. Let X be a space, and let I be the unit interval [0,1] in R, a homotopy of paths in X is a family *gt* : *I* ! *X,* 0≤ *t*≤ 1such that (i) the endpoints *gt* ð Þ¼ 0 *x*<sup>0</sup> and *gt* ð Þ¼ 1 *x*<sup>1</sup> are independent of t and (ii) the associated map *G* : *I* � *I* ! *X* defined by G(s,t) = gt(s) is continuous [15]. Given spaces X and Y with chosen points *x*<sup>0</sup> ∈*X,* and *y*<sup>0</sup> ∈*Y,* the wedge sum X∨Y is the quotient of the disjoint union X∪Y obtained identifying x0 and y0 to a single point [15]. Two spaces X and Y are of the "same homotopy type" if there exist continuous maps *<sup>f</sup>* : *<sup>X</sup>* ! *<sup>Y</sup>* and *<sup>g</sup>* : *<sup>Y</sup>* ! *<sup>X</sup>* such that *<sup>g</sup>* ◦ *<sup>f</sup>* ffi *IX* : *<sup>X</sup>* ! *<sup>X</sup>* and *<sup>f</sup>* ◦ *<sup>g</sup>* ffi *IY* : *<sup>Y</sup>* ! *<sup>Y</sup>* [16]. The fundamental group briefly consists of equivalence classes of homotopic closed paths with the law of composition following one path to another. However, the set of homotopy classes of loops based at the point *x*<sup>0</sup> with the product operation ½ � *f* ½ �¼ *g* ½ � *f* � *g* is called the fundamental group and denoted by *π*1ð Þ *X; x*<sup>0</sup> [4, 17–24]. Over many years, chaos has been shown to be an interesting and even common phenomenon in nature. Chaos has been shown to exist in a wide variety of settings: in fluid dynamics such as Raleigh-Bernard convection, in chemistry such as the Belousov-Zhabotinsky reaction, in nonlinear optics in certain lasers, in celestial mechanics, in electronics in the flutter of an overdriven airplane wing, some models of population dynamics, and likely in meteorology, physiological oscillations such as certain heart rhythms, as well as brain patterns [17, 24–30]. AI algorithms related to adjacency matrices on the operations of the graph are discussed in [31, 32].

*Folding on the Chaotic Graph Operations and Their Fundamental Group DOI: http://dx.doi.org/10.5772/intechopen.88553*
