Linear Representation Of a Graph

In this paper the linear representation of a graph is defined. A linear representation of a graph is a subgroup of $GL(p,\mathbb{R})$, the group of invertible matrices of order $ p $ and real coefficients. It will be demonstrated that every graph admits a linear representation. In this paper, simple and finite graphs will be used, framed in the graphs theory's area


Introduction
It is customary to define or to describe a graph by means of a diagram in which each vertex is represented by a point and each edge e = uv is represented by a line segment or curve joining the points corresponding to u and v.A graph G with vertex set V (G) = {v 1 , v 2 , . . ., v n } and edge set E(G) = {e 1 , e 2 , . . ., e m } can also be described by means of matrices.One such matrix is then n×n adjacency matrix A(G) = (a ij ), where Another matrix is the n × m incidence matrix B(G) = (b ij ), where b ij = 1 if v i and v j are incident 0 otherwise.
In computer science, a graph is an abstract data type that is meant to implement the undirected graph and directed graph.For the representation of graphs, adjacency matrix, incidence matrix and Adjacency list are used.The latter, is a collection of unordered lists used to represent a finite graph.Each list describes the set of neighbours of a vertex in the graph.(See [3]).

Preliminaries
The organization of the following definitions is presented so as this article is selfcontained, for this reason we describe some basic concepts for the understanding of our work.The graphs to be considered will be simple and finite and with a nonempty set of edges.For a graph G, V (G) denotes the set of vertices and E(G) denotes the set of edges.The cardinality of V (G) is called order of G and the cardinality of E(G) is called size of G. Other concepts used in this work and not defined explicitly can be found in the reference [2], [4], [5].
Remark 1.5.S n form a group under the operation of composition, called symmetric group.

Linear Group
In this section, we introduce a fundamental definition for our research.Proof: From (i), and (ii), then Therefore f is an isomorphism.✷

Linear representation of a Graph
Definition 3.1.We will say that M (H) is a linear representation of a graph G, of order p, if only if, M (H) ∼ = Aut(G).

1. 1 .
Simples and finites Graphs Definition 1.1.A graph G is a finite nonempty set of objects called vertices together with a set of unordered pairs of distinct vertices of G called edges.

Remark 3 . 3 .
Note that for the generator characteristic polynomial is: x 4 − 1, and its characteristic values: ±1, ±i.These can be considered as the vertices of a square inscribed in the unit circle in the complex plane.characteristic polynomial is:x 4 −2x 2 +1, and its characteristic values: ±1, each of them of multiplicity two.These can be considered as the generators orthogonal planes in space R 4 .Finally, we have the main result of this work.Theorem 3.4.Every graph admits a linear representation.