Full friendly index set — I

Let G=(V,E) be a graph, a vertex labeling f:V→Z2f:V→Z2 induces an edge labeling f∗:E→Z2f∗:E→Z2 defined by f∗(xy)=f(x)+f(y)f∗(xy)=f(x)+f(y) for each xy∈E. For each, i∈Z2 define vf(i)=|f−1(i)|vf(i)=|f−1(i)| and ef(i)=|f∗−1(i)|ef(i)=|f∗−1(i)|. We call ff friendly if |vf(1)−vf(0)|≤1|vf(1)−vf(0)|≤1. The full friendly index set of GG is the set of all possible values of ef(1)−ef(0)ef(1)−ef(0), where ff is friendly. In this paper, we study the full friendly index sets of some standard graphs such as the complete graph KnKn, the cycle CnCn, fans FmFm and F2,mF2,m and the Cartesian product of P3P3 and PnPn i.e. P3×PnP3×Pn.