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

摘要

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) (mod 2) and f *(xy) = f (x) f (y) for each edge xy∈E. For i∈{0, 1}, let v_f (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 | v_f (1) - v_f (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} f. 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.