On friendly index sets and product-cordial index sets of gear graphs

Abstract

Let G = (V,E) be a simple connected graph. A vertex labeling of f:V→{0,1} of G induces two edge labelings f+, f*:E→{0,1} defined by f+(xy) = f(x)+f(y)(mod2) and f*(xy) = f(x)f(y) for each edge xy ∈ E. For i∈{0,1}, let vf(i) = |{v∈V:f(v) = i}|, ef+(i) = |{e∈E:f+(e) = i}| and ef*(i) = |e∈E:f*(e) = i}|. A labeling f is called friendly if |vf(1)−vf(0)|≤1. The friendly index set and the product-cordial index set of G are defined as the sets {|ef+(0)−ef+(1)|:f is friendly} and {|ef*(0)−ef*(1)|:f is friendly}. In this paper, we completely determine the friendly index sets and product-cordial index sets of gear graphs. We also show that the product-cordial indices of a graph can be obtained from its adjacency matrix.