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⌢ {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}|, e+f(i) = |{e∈E:f+ (e) = i}| and e*f(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 {|e+f(0)-e+f(1)|:f is friendly} and {|e*f(0)-e*f(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. [ABSTRACT FROM AUTHOR]