General-Graph and Inverse-Graph

Unlike other areas in mathematics, graph theory traces its beginning to definite time and place: the problem of the seven bridges of Königsberg, which was solved in 1736 by Leonhard Euler. And in 1752 we find Euler’s Theorem for planer graph. However, after this development, littlie was accomplished in this area for almost a century [1]. There are many physical systems whose performance depends not only on the characteristics of the components but also on the relative locations of the elements. An obvious example is an electrical network. One simple way of displaying a structure of a system is to draw a diagram consisting of points called vertices and line segments called edges which connect these vertices so that such vertices and edges indicate components and relationships between these components. Such a diagram is called linear graph. A graph G is a triple consisting of a vertex-set V(G), an edge-set E(G) and a relation that associated with each edge two vertices called its endpoints. Definition 1. An oriented abstract graph is a pair (V, E) where V is finite non empty set of vertices and E is a set of ordered pairs of distinct elements of E with the property that if (v, w)  E then (w, v)  E where the element (v, w) denote the edge from v to w [1,2]. Definition 2. An empty graph is a graph with no edges [1]. Definition 3. A multiple edges defined as two or more edges joining the same pair of vertices [3-7]. Definition 4. A loop is an edge joining a vertex to itself [3-7]. Definition 5. A simple graph is a graph with no loops or multiple edges [8,9]. Definition 6. A multiple graph is a graph with allows multiple edges and loops [3-7]. Definition 7. A complete graph is a graph in which every two distinct vertices are joined by exactly one edge [5,6,9,10]. Definition 8. A connected graph is a graph that in one piece, where as one which splits in to several pieces is disconnected [6]. Definition 9. Given a graph G, a graph H is called a subgraph of G if the vertices of H are vertices of G and the edges of H are edges of G [1,4,6]. Definition 10. Let v and w be two vertices of a graph. If v and w are joined by an edge, then v and w are said to be adjacent. Also, v and w are said to be incident with e then e is said to be incident with v and w [9]. Definition 11. Let G be a graph without loops, with nvertices labeled 1, 2, 3,···, n. and m edges labeled 1, 2, 3,···, m. The adjacency matrix A(G) is the n × n matrix in which the entry in row i and column j is the number of edges joining the vertices i and j [9]. Definition 12. Let G be a graph without loops, with nvertices labeled 1, 2, 3,···, n and m edges labeled 1, 2, 3,···, m. The incidence matrix I(G) is the nxm matrix in which the entry in row i and column j is l if vertex i is incident with edge j and 0 otherwise [9]. Definition 13. The degree of vertex v in a graph G, written   d v , is the number of edges incident to v, except that each loop at v counts twice[11]. Definition 14. The order of a graph G, written   n G , is the number of vertices in G [11]. Definition 15. The size of a graph G, written   e G , is the number of edges in G [12].


Introduction
Unlike other areas in mathematics, graph theory traces its beginning to definite time and place: the problem of the seven bridges of Königsberg, which was solved in 1736 by Leonhard Euler. And in 1752 we find Euler's Theorem for planer graph. However, after this development, littlie was accomplished in this area for almost a century [1].
There are many physical systems whose performance depends not only on the characteristics of the components but also on the relative locations of the elements. An obvious example is an electrical network. One simple way of displaying a structure of a system is to draw a diagram consisting of points called vertices and line segments called edges which connect these vertices so that such vertices and edges indicate components and relationships between these components. Such a diagram is called linear graph. A graph G is a triple consisting of a vertex-set V(G), an edge-set E(G) and a relation that associated with each edge two vertices called its endpoints. Definition 1. An oriented abstract graph is a pair (V, E) where V is finite non empty set of vertices and E is a set of ordered pairs of distinct elements of E with the pro- Definition 2. An empty graph is a graph with no edges [1]. Definition 3. A multiple edges defined as two or more edges joining the same pair of vertices [3][4][5][6][7]. Definition 4. A loop is an edge joining a vertex to itself [3][4][5][6][7].
Definition 5. A simple graph is a graph with no loops or multiple edges [8,9]. Definition 6. A multiple graph is a graph with allows multiple edges and loops [3][4][5][6][7]. Definition 7. A complete graph is a graph in which every two distinct vertices are joined by exactly one edge [5,6,9,10].
Definition 8. A connected graph is a graph that in one piece, where as one which splits in to several pieces is disconnected [6].
Definition 9. Given a graph G, a graph H is called a subgraph of G if the vertices of H are vertices of G and the edges of H are edges of G [1,4,6].
Definition 10. Let v and w be two vertices of a graph. If v and w are joined by an edge, then v and w are said to be adjacent. Also, v and w are said to be incident with e then e is said to be incident with v and w [9]. Definition 11. Let G be a graph without loops, with nvertices labeled 1, 2, 3,···, n. and m edges labeled 1, 2, 3,···, m. The adjacency matrix A(G) is the n × n matrix in which the entry in row i and column j is the number of edges joining the vertices i and j [9]. Definition 12. Let G be a graph without loops, with nvertices labeled 1, 2, 3,···, n and m edges labeled 1, 2, 3,···, m. The incidence matrix I(G) is the nxm matrix in which the entry in row i and column j is l if vertex i is incident with edge j and 0 otherwise [9].
Definition 13. The degree of vertex v in a graph G, is the number of edges incident to v, except that each loop at v counts twice [11].
Definition 14. The order of a graph G, written   n G , is the number of vertices in G [11].
Definition 15. The size of a graph G, written   e G , is the number of edges in G [12].

The Main Results
In this article we will define a new type of graph which is generalization of the current graph. There exist some problems can not be illustrated by the classical graph, for ex-ample, the relation between students and their studied subjects, as shown in So we want to define a new type of graph illustrate this type of relations.
Before, introducing the definition of the new graph we will define a new types of edges. Definition 1. The original edge is a discrete path between any two vertices i , i , denoted by , such that this path show the existence of the relation.

Now we will introduce types of a general-graph as fo
ion 7. An empty a general-graph is a gener .

llows:
Definit G al graph with vertices but the set E is empty   rtex set can be partitioned into two sets X and Y in such a way that every edge (original edge or inverse edge) of G has one end vertex in X and the other in Y, X and Y are called the partite sets.
Example 3. In figure bl rtite but the third graph is not bipartite because it is not possible to partition the vertices into two such sets.