On Σ and Σ′ labelled graphs

A graph G = (V, E) with delta(G) > 0, where delta(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 is an element of V(G), the sum of all f(nu) where nu is an element of N(u), the neighbourhood of u in G, is a constant independent of u. We call G as a Sigma' labelled graph if there exists a labelling f from V(G) to {1, 2,....vertical bar V(G)vertical bar} such that for all u is an element of V(G), the sum of all f (nu) where nu is an element of N(u) boolean OR {u}, is a constant independent of u. In this paper we give a set of necessary and sufficient condition for the bipartite graph Km, n, m <= n to be a sigma labelled graph. Furthermore, we prove that, the graph G(1) x G(2) with delta(G(i)) 1, vertical bar V(G(i))vertical bar >= 3 for i = 1, 2 is not a sigma labelled graph. Also we prove that every graph is an induced subgraph of a regular Sigma' labelled graph, and some useful properties of Sigma' labelled graph. (C) 2008 Elsevier B.V. All rights reserved.