On Friendly Index Sets and Product-Cordial Index Sets of Gear Graphs

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 -> 1{0,1} defined by f(+)(xy) = f(x)+f(y) (mod 2) and f*(xy) = f(x)f(y) for each edge xy is an element of E. For i is an element of{0,1}, let v(f) (i) = vertical bar v is an element of V : f(v) = i}vertical bar, e(f)(+)(i) = vertical bar e is an element of E: f(+)(e) = i}vertical bar and e(f)*(i) = vertical bar{e is an element of E: f* (e) = i}vertical bar. A labeling f is called friendly if vertical bar v(f) (1) - v(f) (0)vertical bar <= 1. The friendly index set and the product-cordial index set of G are defined as the sets {vertical bar e(f)(+)(0)-e(f)(+)(1)vertical bar: f is friendly} and {vertical bar e(f)*(0) - e(f)*(1)vertical bar: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.