On Tape Graphs

Abstract In this paper we will introduce new type of graphs, when vertices of these graphs are appearance like line and edges of these graphs are appearance like tape or ribbon. We introduce types of representation of the new graph by the adjacent and the incidence matrices and we will discuss their transformations.

(4) The incidence matrix: let G be a graph without loops, with n-vertices labeled 1,2,3,. . .n and m edges labeled 1,2,3,. . .., m. the incidence matrix I(G) is the n x n matrix in which the entry in row i and column j is 1 if vertex i is incident with edge j and 0 otherwise (Gross J.L.,Tucker T.W. , 1987), (Wilson R.J., Watkins J.J. ,1990).
(5) Isometric folding: let M and N be smooth connected Riemannian manifolds of dimensions m and n respectively such that m < n.A map f: M−→ N is said to be an isometric folding of M into N iff for every piecewise geodesic path γ: J → M the induced path f o γ : J → N is a piecewise geodesic and of the same length as γ (Robertson S.A.,1977).
(6) Folding and unfolding of graph: (a) Let f : G → G be a map between any two graphs G and (not necessary to be simple (Giblin P.J., 1977).
(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).
(7) 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
Now we will define and discuss the tape graph and some transformations on this new graph, and that will be represented by matrices.

Definitions
2.1.1The tape graph G is a diagram consisting of a finite non empty set of the elements with "line or curve" shape called "vertices" denoted by V(G) together with elements, with "tape" shape called "edges" denoted by E(G).
The matrix representation of geometric tape graph considers the geometric graph G (v 0 v 1 ) see Fig.

Loop and Multiple Edges
A loop is an edge joining a vertex to itself see Fig.
(2-a).We says that the tape graph has multiple edges if in the tape graph two or more edges joining the same pair of vertices see Fig. (2-b).
We can combine two tape graphs to make a larger graph.If the two tape graphs are A tape graph G is connected if it cannot be expressed as the union of two graphs, and disconnected otherwise.i.e a tape graph G is connected if there is a path in G between any given pair of vertices, otherwise it is disconnected.Every disconnected graph can be expressed as the union of connected graphs, each of which is a component ofG.For example; a graph with three components is shown in Fig ( 4).i.e. a tape graph is connected if and only if there is a path between each pair of vertices.

Tape graph in higher dimension
We can represent the tape graph in higher dimension by matrices as the following:

Folding of geometric tape graph
Theorem 2.2.1 The limit of foldings of tape graph G into itself is a tape graph or a simple graph. Proof: case(1): let f 1 :G→ G ,such that f (tape)=tape then The end of the limits of foldings of a tape graph G into itself coincides with the simple graph.

Proof
Consider the geometric tape graph G (V, E) where =G n which is the usual loop graph see Fig.( 9).Also by matrix: And for incidence: In higher dimensions: And for incidence: The end of limits of the curvature foldings f k n of a tape graph G is the new graph,(circles, tubes). Proof: ).
< Figure 10 > we will arrive to another new graph it is tube graph with circles vertices and tubes edges.
Also by matrix:

Retraction of geometric tape graph
There are many types of retractions of the new tape graph.Consider the geometric tape graph G (V, E) whereV(G) = {V 0 ,V 1 } and E(G)= {e 1 }.
Now let we consider the retractions as: (a) Vertices retractions: r 1 : (G − V 0 ) → G 1 , and r 2 : (G − V 1 ) → G 2 , note that; G 1 is not a graph but lim r 1 n = v 1 is a graph also lim r 2 n is a graph see Fig.                Published by Canadian Center of Science and Education

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Figure 1 > It's adjacent and incidence matrices are:

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Figure 3 >, < Figure 4 > 2.1.4Complete tape graphs A simple 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 > 2.1.5Tree tape graphs A tree tape graph is connected tape graph with only one path between each pair of vertices containing no cycles see Fig. (6).< Figure 6 > It's adjacent and incidence matrices are: (11)  .