A Study on the Inj-Equitable Graph of a Graph

For any graph G, the Inj-equitable graph of a graph G, denoted by ( ) IE G , is the graph with the same vertices as G and for any two adjacent vertices u and v in ( ) IE G , ( ) ( ) deg deg 1 in in u v − ≤ , where for any vertex ( ) w V G ∈ , ( ) ( ) ( ) { } deg : in w w V N w N w φ ′ ′ = ∈ ≠  . In this paper, Inj-equitable graphs of some graphs are obtained, and some properties and results are established. Moreover, complete Inj-equitable graph and the Inj-equitable graph are defined.

. In this paper, Inj-equitable graphs of some graphs are obtained, and some properties and results are established.Moreover, complete Inj-equitable graph and the Inj-equitable graph are defined.

Introduction
We consider only finite undirected graphs ( ) without loops and multiple edges.X will denote the subgraph of G induced by a set of vertices X.For any vertex

( )
V G copies of H and by joining each vertex of the i-th copy of H to the i-th vertex of G, where ( ) { , , , : . For more terminologies and notations, we refer the reader to [1] [2] [3] and [4].
The common neighborhood graph (congraph) of a graph G, denoted by , in which two vertices are adjacent if and only if they have at least one common neighbor in the graph G [5].The equitable graph of a graph G, denoted by e G , is the graph with vertex set ( ) V G and two vertices u and v are adjacent if and only if

IE-Graph of a Graph
The Inj-equitable dominating sets on graphs which introduced in [7] motivated us to define two new graphs: the IE-graph of a graph and the IE-graph.In this section, we define the Inj-equitable graph of a graph and study some properties of this graph.Also, the injective equitable graph of some graph's families are found.

Definition 1 Let
( ) Let G be a graph as in Figure 1.Then, The Inj-equitable graph of some known graphs are given in the following observation: We will generalize Proposition 4 in the following result.Proposition 5 For any multipartite graph  Proof.Let G be a multipartite graph Bi-star graph is the graph obtained by joining the apex vertices of two copies of star 1,n K .
Proposition 6 For any bi-star graph ( ) be a bi-star graph labeling as in Figure 2. Therefore, ( ) as in Figure 3, where  be any end vertex in the pendant edge.i u and , 1, 2, , i u i s ′ =  be any end vertex and internal vertex respectively in the pendant path.Then, ( ) There are three cases: Case 1. Suppose that  ( ) ( ) , , , such that there are i s common vertices between i n K and Proposition 9 Let G be a graph such that Then there are two cases: Therefore, for any two vertices u and v, ( ) ( ) where the subscripts are integers modulo m, 5 m ≥ and , , , , , , , , , , ) where n is the number of vertices in G.
Proof.Let G be a k-regular graph with n vertices and 2 δ ≥ .Suppose that are the vertex sets of G and m K , respectively.
Therefore, for 1, 2, , and for . Similarly, we can prove if G is ( ) Proposition 18 For any cycle graph n C , ( ) Proof.Let i u be any vertex on the cycle n C and i v be any vertex outside the = be a graph.A subset D of V is called degree Inj-equitable set if the difference between the injective degree of any two vertices in D less then or equal one.The maximum cardinality of a degree Inj-equitable set in G is called degree Inj-equitable number of G and is denoted by the minimum cardinality of a maximal degree Inj-equitable set is called the lower degree Inj-equitable number of G and is denoted by Observation 20 For any integer  ,  G V E = be a graph.The intersection graph of maximal degree Inj-equitable sets denoted by in G is defined on the family of all maximal degree Inj-equitable sets of G and has the property that any two vertices are adjacent if their intersection is not empty.

is a maximal degree Inj-equitable if and only if
Theorem 22 For any graph G, ( ) Similarly, we can show that The following proposition can be proved straightforward.
Proposition 23 For any graph G, Theorem 24 Let G be any graph.The number of edges in ( ) IE G is given by: ( ) ( ) where ( ) S S − edges.But the edges in Theorem 25 Let G be any graph.Then the following are equivalent: (ii) The distinct sequence of the Inj-equitable degrees are  ( ) any vertices in G such that 1 v adjacent to 2 v and 2 v adjacent to 3 v .Suppose that ( )

Conclusion
In this paper, we introduced the injective equitable graph of a graph ( ) IE G and the injective equitable graph IE-graph.We defined these graphs and presented some of their properties.Also, we found the Inj-equitable graph of some graph's families.The degree injective equitable set, the degree injective equitable number of a graph and the lower degree injective equitable number are defined.Relations on those parameters in terms of maximum vertex degree, minimum vertex degree, diameter and clique number of the injective equitable graph are established.The connectedness of ( ) IE G and relations between the ( )

( )
Con G and e G are studied.In the last of this paper, the sufficient condition for a connected graph G to be IE-graph is presented.
vertex degree of G respectively.The distance ( ) , d u v between any two vertices u and v in a graph G is the number of the edges in a shortest path.The eccentricity of a vertex u in a connected graph G is diameter of G is the value of the greatest eccentricity, and the radius of G is the value of the smallest eccentricity.The clique number of G, denoted by ( ) G ω , is the order of the maximal complete subgraph of G.The Inj-neighborhood of a vertex ( ) u V G ∈ denoted by ( ) in N u is defined as is defined as the graph with the same vertices as G and two vertices u and v are adjacent in ( )
Figure 4.Then, . Therefore, all the vetrices have the same Inj-degree.Hence, ≅ × .Then we have 2m vertices of injective degree 5, 2m vertices of injective degree 7 and

Figure 4
Figure 4. Generalized Petersen graph ) ie D G is the order of the maximum complete subgraph in ( ) IE G .Therefore, for any vetrex in the maximum complete subgraph has degree ( )
is called injective degree of the vertex u and is denoted by in N u The intersection graph of maximal degree Inj-equitable sets in G is a path.
then 1Hence the graph H has different injective degrees which contradicts that any graph with n vertices has at least two vertices of the same Inj-degree.Therefore, Proof.Suppose, to the contrary, that G is a cycle such that G is IE-graph.Therefore, there exist a graph H such that Hence the graph H has different injective degrees which contradicts that any graph with n vertices has at least two vertices of the same Inj-degree.Therefore, any cycle n C is not IE-graph.Lemma 34 Any bipartite graph is not IE-graph.Proof.Suppose, to the contrary, that G is a bipartite graph such that G is IE-graph.Therefore, there exists a graph H such that and 3 v are adjacent in G. So, G contains an odd cycle which contradicts that G is a bipartite graph.Therefore, , then 2 v and 3 v are adjacent in G. So, G contains an odd cycle which contradicts that G is a bipartite graph.Therefore, .Hence the graph H has different injective degrees which contradicts that any graph with n vertices has at least two vertices of the same Inj-degree.Therefore, any bipartite graph is not IE-graph.Theorem 35 A connected graph G is IE-graph if G is a chain of complete graphs.Proof.Suppose that G is a connected IE-graph.Then there exists a graph H v