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.
- chaotic graph
- edge graph
- chaotic folding
- limit folding fundamental group
- 2010 Mathematics Subject Classification: 51H20
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) ≠ 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 A null graph is a graph containing no edges; the null graph with n-vertices is denoted by A cycle graph is a graph consisting of a single cycle, the cycle graph with n-vertices is denoted by The path graph is a graph consisting of a single path; the path graph with n-vertices is denoted by Pn [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. Let G and H be two graphs. A function is a homomorphism from G to H if it preserves edges, that is, if for every edge [12, 13]. A core is a graph which does not retract to a proper subgraph. Any graph is homomorphically equivalent to a unique core .
The folding is a continuous function such that for each and for each . 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 such that (i) the endpoints and are independent of t and (ii) the associated map defined by G(s,t) = gt(s) is continuous . Given spaces X and Y with chosen points and the wedge sum X∨Y is the quotient of the disjoint union X∪Y obtained identifying x0 and y0 to a single point . Two spaces X and Y are of the “same homotopy type” if there exist continuous maps and such that and . 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 with the product operation is called the fundamental group and denoted by [4, 17, 18, 19, 20, 21, 22, 23, 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, 25, 26, 27, 28, 29, 30]. AI algorithms related to adjacency matrices on the operations of the graph are discussed in [31, 32].
2. The main results
First, we will introduce the following:
Since and , it follows that , and so reduces the rank of Also, if and , we have, ; thus reduces the rank of Moreover, by continuing with this procedure if , then . Hence, reduces the rank of .
has no chaotic cycles, then clearly is a chaotic edge and .
3. Some applications
A polymer is composed of many repeating units called monomers. Starch, cellulose, and proteins are natural polymers. Nylon and polyethylene are synthetic polymers. Polymerization is the process of joining monomers. Polymers may be formed by addition polymerization; furthermore, one essential advance likewise polymerization is mix as in Figure 4, which happens when the polymer’s development is halted by free electrons from two developing chains that join and frame a solitary chain. The accompanying chart portrays mix, with the image (R) speaking to whatever remains of the chain.
Chemical nature of enzymes, all known catalysts are proteins. They are high atomic weight mixes made up primarily of chains of amino acids connected together by peptide bonds as in Figure 5.
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.
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 new operation of a graph by using the adjacency matrices.