Product-cordial index and friendly index of regular graphs

Let g = (v;e) be a connected simple graph. a labeling f : v rightnarrow z2 induces two edge labelings f+; f* : e rightnarrow z2 defined by f+(xy) = f(x)+f(y) and f*(xy) = f(x)f(y) for each xy ε e. for i ε z2, let vf (i) = jf-1(i)j, ef+(i) = j(f+)-1(i)j and ef* (i) = j(f*)-1(i)j. a labeling f is called friendly if vert vf (1)-vf (0) vert leq 1. for a friendly labeling f of a graph g, the friendly index of g under f is defined by if +(g) = ef+(1) - ef+(0). the set {fi+f (g) vert f is a friendly labeling ofg} is called the full friendly index set of g. also, the product-cordial index of g under f is defined by i*f (g) = ef* (1) - ef* (0). the set fi*f (g) vert f is a friendly labeling ofgg is called the full product-cordial index set of g. in this paper, we nd a relation between the friendly index and the product-cordial index of a regular graph. as applications, we will determine the full product-cordial index sets of torus graphs which was asked by kwong, lee and ng in 2010; and those of cycles