Minimum Rank of Graphs Powers Family

In this paper we study the relationship between minimum rank of graph G and the minimum rank of graph j G for some families of special graph G, where j G is the jth power of graph G.


Introduction
A graph is a pair , where V is the set of vertices (usually or a subset thereof) and E is the set of edges (an edge is a two-element subset of vertices); what we call a graph is sometimes called a simple undirected graph.In this paper each graph is finite and has nonempty vertex set.The order of a graph G, denoted G , is the number of vertices of G.A path is a graph A cycle is a graph The length of a path or cycle is the number of its edges.A complete graph is a graph


if the vertex set V can be partitioned into two nonempty subsets U and W, such that every edge of E has one endpoint in U and one in W. A complete bipartite graph is a bipartite graph The line graph of a graph denoted is the graph having vertex set E, with two vertices in adjacent if and only if the corresponding edges share an endpoint in G. Since we require a graph to have a nonempty set of vertices, the line graph is defined only for a graph G that has at least one edge.

 
The corona of G with H, denoted , is the graph of order obtained by taking one copy of G and G copies of H, and joining all the vertices in the ith copy of H to the ith vertex of G. See Figures 6, 7 for a picture of 6 ). Definition 1.1 The j th power of a graph G is a graph with the same set of vertices as G and an edge between two vertices if there is a path of length at most j between them.
Definition 1.2 For such a matrix, the graph of A, denoted   G A , is the graph with vertices and edges the corank of A is the nullity of A and the maximum nullity (or maximum corank) G (over R) is defined to be of a graph More generally, the minimum rank of a simple graph G is defined to be the smallest possible rank o all symmetric real matrices whose ijth entry (for i j ver  ) is nonzero whenever   , i j is an edge in G o ot of an eigenvalue among the same family of ma and is zer trices [3].
herwise [1,2].The solution to the minimum rank problem is equivalent to the determination of the maximum multiplicity

  Z G
Here we introduce the graph parameter   Z G as the minimum size of a zero forcing set from [1].The zero forcing number is a useful tool for determining the minimum rank of structured families of graphs and small graphs [4].
Definition 2.1 Color-change rule:  If G is a graph with each vertex colored either white or black, u is a black vertex of G, and exactly one neighbor v of u is white, then change the color of v to black. Given a coloring of G, the derived coloring is the result of applying the color-change rule until no more changes are possible. A zero forcing set for a graph G is a subset of vertices Z such that if initially the vertices in Z are colored black and the remaining vertices are colored white, the derived coloring of G is all black.
For example, an endpoint of a path is a zero forcing set for the path.In a cycle, any set of two adjacent vertices is a zero forcing set.
Corollary 2.2 [1,5] Let be a graph and let , and thus The Colin deVerdiere-type parameter  can be useful in computing minimum rank or maximum nullity (over the real numbers).A symmetric real matrix M is said to satisfy the Strong Arnold Hypothesis provided there does not exist a nonzero symmetric matrix X satisfying: 0. = 0. = 0.

MX M X I X
where denotes the Hadamard (entrywise) product and I is the identity matrix.For a graph G, is the maximum nullity among matrices A S G  that satisfy the Strong Arnold Hypothesis.
It follows that .
A contraction of G is obtained by identifying two adjacent vertices of G, and suppressing any loops or multiple edges that arise in this process.A minor of G arises by performing a series of deletions of edges, deletions of isolated vertices, and/or contractions of edges.A graph parameter  is minor monotone if for any minor G of G, The parameter  was introduced in [6], where it was shown that  is minor monotone.It was also established that  = Other possible bounds for minimum rank derived from certain easy to compute parameters of the graph were considered, leading to an investigation of the connection between minimum degree of a vertex,  , G  and minimum rank [8].
Corollary 2. 4 For any graph G and infinite field F,

Main Results
the minimum rank of graph In here we calculate Theo .
. (The minimum ra ost 7 a nks of all graphs of order at m re available in the spreadsheet [9]).
n n n K C P 3) Any tree T. 4) Some families of special graphs. Theorem.

es of gr
By this Proposition we have following Theorem 3.3 For each of the following famili Peterson Graph 9  (see also [10]).
Proof.(Figure 2) C any vertex u is adjacent to 2j vertices, then On the other hand, .Then and finally .
x, then vertices.

C K
and hence Proposition 3.11   For this purpose we obtained Zero forcing set and graph parameter   Z G and the parameter   because if we star oloring from u, this vertex at least is adjacent with j vertices.The vertex u with its 1 j  adjacent vertices are coloring.The other vertices a re coloring since they are adjacent to coloring vertices, and the number of coloring vertices is j, therefore we have   =

3 , 4 , 5 )
In the jth power of graph an external vertex is adjacent to

3 4 j
If we start to co ng of external verte vertices are coloring, which lori from "color-change rule".We continue the process until all vertices are colored on the internal cycle.Finally Theorem 3.10 For all 2 u are sam cent verti f em is different.O e r cycle has colored and the anothe the rtex that is located on th ycle colored n internal cycle reach to complete graph.With contraction of 2 1 j  vertices to 2 3 j power  vertices of this graph we reach to complete graph of order

Table 1 . Su ar
mm y of minimum rank results established in this paper.G