On and labelled graphs

A graph G = ( V , E ) with δ ( G ) > 0 , where δ ( G ) is the minimum degree among the vertices of G , is said to be a sigma labelled graph if there exists a labelling f from V ( G ) to { 1 , 2 , … , | V ( G ) | } such that for all u ∈ V ( G ) , the sum of all f ( v ) where v ∈ N ( u ) , the neighbourhood of u in G , is a constant independent of u . We call G as a ∑ ′ labelled graph if there exists a labelling f from V ( G ) to { 1 , 2 , … , | V ( G ) | } such that for all u ∈ V ( G ) , the sum of all f ( v ) where v ∈ N ( u ) ⋃ { u } , is a constant independent of u . In this paper we give a set of necessary and sufficient condition for the bipartite graph K m , n , m ≤ n to be a sigma labelled graph. Furthermore, we prove that, the graph G 1 × G 2 with δ ( G i ) = 1 , | V ( G i ) | ≥ 3 for i = 1 , 2 is not a sigma labelled graph. Also we prove that every graph is an induced subgraph of a regular ∑ ′ labelled graph, and some useful properties of ∑ ′ labelled graph.