Non-neighbour irregular graphs

: A graph G is said to be non-neighbour irregular graph if no two nonadjacent vertices of G have same degree. This paper suggests the methods of construction of non-neighbour irregular graphs. This paper also includes a few properties possessed by these non-neighbour irregular graphs.


Introduction
Throughout this paper we consider only undirected, finite and simple graphs. Let G be such graph with vertex set ) is the number of vertices adjacent to v and is denoted by K , is a complete bipartite graph on q p  vertices and n P is a path with n vertices. If G and H are graphs with the property that the identification of any vertices of G with an arbitrary vertex of H results in a unique graph (up to isomorphism), then we write G·H for this graph. Let  x (  x ) denote the least (greatest) integer greater (less) than or equal to x . Defining a new graph from a given graph, by using incident relationship between vertices and edges and adjacency relation between two vertices or two edges is known as graph transformation. Sometimes graph transformation is referred as a graph valued function. Perhaps one of the first graph transformation of a graph G is its complement denoted by G , which is a graph with vertex set V(G) in which two vertices are adjacent if and only if they are nonadjacent in G and ) , where n is the number of vertices of G . A selfcomplementary graph is isomorphic with its complement. Notations and terminology that we do not define here can be found in [4,5].
Regular graphs are those graphs for which each vertex has same degree. There are plenty of regular graphs, for example, complete graph. The problem arises when a graph is not regular. If it is irregular how much of irregularity is thrust upon its vertices? In this connection three new concepts called highly irregular graphs, k -neighbourhood graphs and neighbourly irregular graphs [1, 2, 3] have evolved. A connected graph G is said to be highly irregular if each neighbour of any vertex has different degree. A connected graph G is said to be a k-neighbourhood regular graph if each of its vertices is adjacent to exactly k -vertices of the same degree. A connected graph G is said to be neighbourly irregular graph abbreviated as NI graph if no two adjacent vertices of G have the same degree.
Inspired by these three definitions we define the concept of non-neighbour irregular graphs abbreviated as NNI graphs.

A graph
G is said to be non-neighbour irregular if no two nonadjacent vertices of G have the same degree. For example, the graphs shown in Fig. 1 are NNI graphs.
This concept is helpful to study the NI for the graph transformation G G  . Note that the graph which is not NI is not necessarily a NNI graph. For example consider the graph 1 G shown in Fig. 2 is not NI, but 1 G is NNI. The graph 2 G shown in Fig. 2 Proof. The required connected NNI graph is constructed as follows. The n vertices are partitioned into k sets. The first set consists of 1 n vertices . Therefore the order of NNI graph is 1   k n . Since each ′ s forms complete graph and degree of vertex in each The size of NNI= degv Order of (2,3,4)

Corollary 2.2 The maximum size of such a connected NNI graph of order
Proof. The required disconnected NNI graph is constructed as follows. The n vertices are partitioned into k sets. The first set consists of 1 n vertices Order of (1,3,4) Size of ( 3 Properties of NNI graphs

Clique graphs.
A clique of a graph is a maximal complete subgraph of G . The clique graph of G is the intersection graph of all cliques of G . Consider the connected NNI graph represented by k sets of distinct partitions of n . By construction of this connected NNI graph each clique of it is of order i n . Also if r n is the maximum number in the partition of n , then the intersection of the clique of order r n and the remaining 1  k cliques is nonempty. This shows that the clique graph is a star graph of order k and size 1  k . Summarizing, therefore, we get

Edge covering number.
A set of edges which covers all the vertices of a graph G is called a edge cover. The smallest number of edges in any edge cover of G is called its edge covering number and it is denoted by

Theorem 3.5
The edge covering number of connected

Vertex independence number.
The set of vertices in G is independent if no two of them are adjacent. The largest number of vertices in such a set is called vertex independence number of G and is denoted by   BMSA Volume 15