**2. The main results**

First, we will introduce the following:

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

**Case 2** (2) *e*1*, e*2*, e*3*,* … *represent different physical properties; for example, e*<sup>1</sup> *represents density, e*<sup>2</sup> *represents hardness, e*<sup>3</sup> *represents magnetic fields, and so on*.

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

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

**Definition 4.** *A chaotic graph H is called a chaotic subgraph of a chaotic graph G if V H* ⊆*V G and E H* ⊆*E G* .

**Definition 5.** *Let G and H be two chaotic graphs. A function φ* : *V G* ! *<sup>V</sup> <sup>H</sup> is chaotic homomorphism from G to H if it preserves chaotic edges, that is, if for any chaotic edge* ½ � *u; v of G,* ½ � *φ*ð Þ *u ; φ*ð Þ*v is a chaotic edge of H*.

**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 every chaotic vertex x of H.*

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

**Theorem 1.** *Let G*<sup>1</sup> *and G*<sup>2</sup> *be two chaotic connected graphs. Then π*<sup>1</sup> G1 ⊻ G2 <sup>¼</sup> *<sup>π</sup>*<sup>1</sup> G1 ∗ *π*<sup>1</sup> G2 .

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

*π*<sup>1</sup> G1 ⊻ G2 ≈*π*<sup>1</sup> G1 ∗ *π*<sup>1</sup> G2 *:* Hence, *π*<sup>1</sup> G1 ⊻ G2 <sup>¼</sup> *<sup>π</sup>*<sup>1</sup> G1 ∗ *π*<sup>1</sup> G2 .

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

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

*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

pendent 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

ð Þ¼ 0 *x*<sup>0</sup> and *gt*

ð Þ¼ 1 *x*<sup>1</sup> are inde-

The folding is a continuous function *f* : *G* ! *H* such that for each

let I be the unit interval [0,1] in R, a homotopy of paths in X is a family

equivalent to a unique core [7].

*Königsberg seven-bridge problem.*

*Functional Calculus*

**Figure 1.**

**52**

*gt* : *I* ! *X,* 0≤ *t*≤ 1such that (i) the endpoints *gt*

operations of the graph are discussed in [31, 32].

$$\begin{aligned} &\text{1. 2607em} \ \text{1. } \text{g} \ \text{ } \Box\_1 \text{ ama } \Box\_2 \text{ are concave} \text{ como} \\ &\text{1. } \overline{\pi}\_1 \left( \lim\_{n \to \infty} \overline{f}\_n \left( \overline{\text{G}}\_1 \underline{\vee} \, \overline{\text{G}}\_2 \right) \right) = \overline{\pi}\_1 \left( \lim\_{n \to \infty} \overline{f}\_n \left( \overline{\text{G}}\_1 \right) \right) \ast \overline{\pi}\_1 \left( \lim\_{n \to \infty} \overline{f}\_n \left( \overline{\text{G}}\_2 \right) \right). \end{aligned}$$

**Proof.** If *G*<sup>1</sup> and *G*<sup>2</sup> are chaotic connected and not chaotic cores graphs, then we get the following chaotic induced graphs lim*n*!<sup>∞</sup> *f <sup>n</sup> G*<sup>1</sup> ⊻ *G*<sup>2</sup> *,* lim*n*!<sup>∞</sup> *<sup>f</sup> <sup>n</sup> <sup>G</sup>*<sup>1</sup> *,* lim*n*!<sup>∞</sup> *f <sup>n</sup> G*<sup>2</sup> , and each of them are isomorphic to *k*2. Since *k*<sup>2</sup> ≈*k*<sup>2</sup> ⊻ *k*<sup>2</sup> it follows that lim*n*!<sup>∞</sup> *f <sup>n</sup> G*<sup>1</sup> ⊻ *G*<sup>2</sup> <sup>¼</sup> lim*n*!<sup>∞</sup> *<sup>f</sup> <sup>n</sup> <sup>G</sup>*<sup>1</sup> <sup>⊻</sup> lim*n*!<sup>∞</sup> *<sup>f</sup> <sup>n</sup> <sup>G</sup>*<sup>2</sup> and *π*<sup>1</sup> lim*n*!<sup>∞</sup> *f <sup>n</sup> G*<sup>1</sup> ⊻ *G*<sup>2</sup> <sup>=</sup> *<sup>π</sup>*<sup>1</sup> lim*n*!<sup>∞</sup> *<sup>f</sup> <sup>n</sup> <sup>G</sup>*<sup>1</sup> <sup>∗</sup> *<sup>π</sup>*<sup>1</sup> lim*n*!<sup>∞</sup> *<sup>f</sup> <sup>n</sup> <sup>G</sup>*<sup>2</sup> *:*

## **3. Some applications**


iii. There are two types of the subunit structure of ribosomes as in **Figure 6**

rRNA to form a new type of ribosomes.

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

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

new operation of a graph by using the adjacency matrices.

\*Address all correspondence to: m\_abusaleem@bau.edu.jo

provided the original work is properly cited.

**4. Conclusion**

*Prokaryotic ribosome components.*

**Figure 6.**

**Author details**

Jordan

**57**

Mohammed Abu Saleem

which is represented by the different connected types of protein subunit and

In this chapter, the fundamental group of the limit chaotic foldings on chaotic connected edge graphs is deduced. Also, we can deduce some algorithms from a

Department of Mathematics, Faculty of Science, Al-Balqa Applied University, Salt,

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

**Figure 5.** *Typical amino acids.*

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

#### **Figure 6.** *Prokaryotic ribosome components.*

iii. There are two types of the subunit structure of ribosomes as in **Figure 6** which is represented by the different connected types of protein subunit and rRNA to form a new type of ribosomes.
