On ∑∑ and ∑′∑′ labelled graphs

A graph G=(V,E)G=(V,E) with δ(G)>0δ(G)>0, where δ(G)δ(G) is the minimum degree among the vertices of GG, is said to be a sigma labelled graph   if there exists a labelling ff from V(G)V(G) to {1,2,…,|V(G)|}{1,2,…,|V(G)|} such that for all u∈V(G)u∈V(G), the sum of all f(v)f(v) where v∈N(u)v∈N(u), the neighbourhood of uu in GG, is a constant independent of uu. We call GG as a ∑′∑′labelled graph   if there exists a labelling ff from V(G)V(G) to {1,2,…,|V(G)|}{1,2,…,|V(G)|} such that for all u∈V(G)u∈V(G), the sum of all f(v)f(v) where v∈N(u)⋃{u}v∈N(u)⋃{u}, is a constant independent of uu. In this paper we give a set of necessary and sufficient condition for the bipartite graph Km,n, m≤nm≤n to be a sigma labelled graph. Furthermore, we prove that, the graph G1×G2G1×G2 with δ(Gi)=1,|V(Gi)|≥3δ(Gi)=1,|V(Gi)|≥3 for i=1,2i=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.