Chaotic Tape Graphs

Received: December 15, 2010 Accepted: December 23, 2010 doi:10.5539/jmr.v3n2p182 Abstract In this paper we will introduce new type of chaotic graphs, when vertices of these graphs are appearance like line and edges of these graphs are appearance like tape or ribbon and when these graphs carries physical characters . The representation of the chaotic graphs by matrices will be obtained. The transformations of the chaotic graph into itself and into another graphs will be discussed.

(b) Let g : G → G be a map between any two graphs G and (not necessary to be simple) such that if (u, v) ∈ G , (g, (u), g(v)) ∈ G .Then g is called a "topological unfolding" of toG provided that d(g(u), g(v)) > d (u, v) (El-Ghoul M., 2007).

Retracts:
A subset A of a topological space X is called a "retract" of A if there exists a continuous map r : X → A (called a retraction) such that r(a) = a∀a ∈ A, where A is closed and X is open.In other words, a retraction is a continuous map of a space onto a subspace leaving each point of the subspace fixed (El-Ghoul M.,El-Ahmady A., Rafat H., 2004).

Main Results
In the following, we will define the chaotic graph as the graph which carries many physical characters.The representation of these chaotic graph by matrices will be defined.Also some transformations of this chaotic graph into itself will be acheived.

Definitions
The geometric tape graph G is a diagram consisting of a finite non empty set of the elements with "line or curve" shape called "vertices" dented by V(G) together with elements, with "tape" shape called "edges" denoted by E(G).When that geometric tape graph carrying physical characters we called "chaotic tape graph" denoted by G h .And there are two types of chaotic of tape graph: a-Chaotic of one face: A physical characters carries on one face of geometric tape graph like painting, printing,glue ...etc.b-Chaotic of two faces: A physical characters carrying on two faces of geometric tape graph like color, electricity, magnetic filed...etc.

1-The matrix representation of geometric tape graph
Considers the geometric graph G (v 0 v 1 ).See

2-The matrix representation of chaotic tape graph
Considers the chaotic graph G h (v 0 0h v 1 0h ), this graph consists of the geometric edges e 1 0h and smooth chaotic edges e 1 ih , i = 1, 2, 3, ..., ∞, of one maximum point see Fig.
These matrcies represent the chaotic of one face.

Complete chaotic of tape graphs:
A simple geometric tape graph in which each pair of distinct vertices are adjacent is a complete tape graph.We denote the complete tape graph on n vertices k, k has n (n-1)/2 edges see Fig. (5).

< Figure 5 >
And Its adjacent and incidence matrices are: We can represent chaotic of complete tape graph see Fig. (6).And It's adjacent and incidence matrices are:  < Figure 8 > The adjacent and incidence matrices of complete chaotic of two faces are: Tree tape graphs: A tree tape graph is connected tape graph with only one path between each pair of vertices containing no cycles see Fig.

Folding of chaotic tape graph
There are two type of folding, we will illustrate their in the following theorems: Theorem 2.2.1: The folding of the geometric tape graph induce the same type of folding of all chaotics.
Lemma 2.2.1: The limit of foldings of the geometric tape graph induce the limit of all foldings of chaotics.

Proof:
Consider the chaotic tape graph G h which consists of the geometric edge e 1 0h and the chaotic edges e 1 ih , i = 1, 2, ..∞.Let f : G h → G h be a folding edge e 1 0h and the chaotic edges e 1 ih such that f 1 : And for incidence: Theorem 2.2.2: The folding for the pure chaotics edges of a chaotic tape graph into itself does not induce a folding for the geometric edge.

Proof:
Consider the chaotic tape graph G h which consists of the geometric edge e 1 0h and the chaotic edges e 1 ih , i = 1, 2, ..∞.Let f : G h → G h be a folding edge such that f (G h ) = G h and f (e 1 ih ) = e 1 jh where i j, j = 1, 2, ..∞ then at each folding the number of the chaotic edges is reduced by 1 i.e. f (e 1 12..i.. j..∞h ) = e 1 12.. j..∞h .By repeating this folding many times then the chaotic tape graph G h becomes a 1-chaotic on its geometric edge e 1 0h , then this type of folding does not induce a folding for the geometric edge.
Lemma 2.2.2: The limit of foldings for the pure chaotics edges of a chaotic tape graph into itself have not any effect on the geometric edge.

Proof:
Let f : G h → G h be a folding edge such that f (G h ) = G h and f (e 1 ih ) = e 1 ih where e 1 ih is a different density from e 1 ih ∀ i.This different may be in the length or in the physical characters (reduce or increase), all these types of folding for the pure chaotics edges have not any effect on the geometric edge i.e. e 1 0h still invariant see Fig.
And for incidence: Where 1 012 f 012 f means that we make a folding for the e 1 2h chaotic edge.Lemma 2.2.3: The unfolding of the chaotics edges of tape graph into itself decreases the chaotics dgree.

Retraction of chaotic tape graph
There are two types of retractions for chaotics tape graph: (a) Geometric retraction: → G 1 h for the geometric edge as we mentioned, consequently the chaotic edges will be retracted as the geometric edge, then we get a chaotic vertex at v 0 0h or v 1 0h , the adjacent and the incidence matrices are: Making a retraction for the chaotic edges e 1 ih ,i = 1, 2, ...∞h of the chaotic graph G h , by removing an interior point which leads to reduce the density of the physical character or vanishing the chaotic edge which we make a retraction for it, and that has no effect on the geometric edge or on the other chaotic edges.So by the retraction r : G h − e 1 ih → G 3 h ,we can reduce the density of the chaotic edges.The adjacent and the incidence matrices are: Where 1 01 r 2..∞h means that we make a retraction for the chaotic edge e 1 1h this retraction is equivalent to the folding of the chaotic edge e 1 1h on e 1 2h and that has no effect on the geometric edge or the other chaotic edges.Theorem 2.3.1:Any type of retractions for the geometric tape graph of a chaotic tape graph into itself induces the same retraction for chaotic edges.

proof:
Let r : G h − {v} → G h be a retraction for the geometric edge of the chaotic tape graph G h , v is either a vertex (v 0 0h or v 0 1h ) or an interior point v 0h on this edge.So we get the following retraction's cases: , then there is an induced sequence of retractions of it's chaotic , then there is an induced sequence of retractions of it's chaotic Any type of retractions for the pure chaotic system of a chaotic tape graph into itself does not induce any retraction for the geometric edge.

proof:
Let r i : G h − {v ih } → G h be a retraction for the pure chaotic edges of the chaotic tape graph G h , v ih is either a chaotic vertex (v 0 ih or v 0 ih ) or an interior chaotic point v ih on the chaotic edges.So we get the following retraction's cases: ih , then there are induced sequences of retractions of their chaotics such that: These retraction's cases do not induce a retraction for the geometric edge i.e.
.∞ are defined as r 2 (e 1 ih − v 1 ih ) = v 0 ih , then there are induced sequences of retractions of their chaotics such that: These retraction's cases do not induce a retraction for the geometric edge i.e. r 2 i : (e ih then there are the induced sequences of retractions of their chaotics such that: These retraction's cases do not induce a retraction for the geometric edge i.e.r 3 i : (e 1 ih − v ih ) r 3 (e 1 0h − v 0h ).Lemma 2.3.1: Any retraction for some chaotic edges, doesn't induces a retraction for the remaining chaotic edges.The limit of sequances of deformations of a chaotic tape graph is one of these deformations.
Theorem 2.4.2: Any type of deformation for the pure chaotic system of a chaotic tape graph into itself does not induce any deformation for the geometric edge.

proof:
Consider the chaotic tape graph G h which consists of the geometric edge e 1 0h and the chaotic edges e 1 ih , i = 1, 2, ..∞.Let d : G h → G h be a deforming pure chaotic edges such that d(G h ) = G h and d(e 1 ih ) = e 1 ih where e 1 ih is a different form e 1 ih ∀ i.This different in the physical characters (reduce or increase), all these types of deforming for the pure chaotics edges have not any effect on the geometric edge i.e. e 1 0h still invariant, see Fig. (1) A ribbon of cassette recorder its geometric tape graph and chaotic tape graph, it has physical characters (colors and voices).
(2) A ribbon of camera film its geometric tape graph and chaotic tape graph, it has physical characters (colors and reflection).
(3) A ribbon of cassette video recorder its geometric tape graph and chaotic tape graph, it has physical characters (colors, voices and dynamics).
(4) A magnetic shapes which we put it on a fridge.
(5) Changing voices or colors in the screen of the television.
(6) Crash a plane induces to jamming of wireless set of it.
(7) Jamming of wireless set does not induce any deformation to the set itself.
(8) The rainbow is a chaotic tape graph.