A Counter Example For Neighbourhood Number Less Than Edge Covering Number Of a Graph

The open neighbourhood N(v) of a vertex v 2 V; is the set of all vertices adjacent to v. Then N[v] = N(v)[fvg is called the enclave of v. We say that a vertex v 2 V, n-covers an edge x ∈ X if x ∈ {N[v]i}, the subgraph induced by the set N[v]. The n-covering number pn(G) introduced by Sampathkumar and Neeralagi [18] is the minimum number of vertices needed to n-cover all the edges of G. In this paper one of the results proved in [18] is disproved by exhibiting an infinite class of graphs as counter example. Further, an expression for number of triangles in any graph is established. In addition, the properties of clique regular graphs has been studied.