On product-cordial index sets and friendly index sets of 2-regular graphs and generalized wheels

Abstract

A vertex labeling f: V → ℤ2 of a simple graph G = (V, E) induces two edge labelings f+, f*: E → ℤ2 defined by f+(uυ) = f(u) + f(υ) and f*(uυ) = f(u)f(υ). For each i ∈ ℤ2, let υf(i) = |{υ ∈ V: f(υ) = i}|, ef+(i) = |{e ∈ E: f+(e) = i}| and e*f(i) = |{e ∈ E: f*(e) = i}|. We call f friendly if |υf(0) − υf(1)| ≤ 1. The friendly index set and the product-cordial index set of G are defined as the sets {|ef+f(0) − ef+(1)|: f is friendly} and {|e*f(0) − e*f(1)|: f is friendly}. In this paper we study and determine the connection between the friendly index sets and product-cordial index sets of 2-regular graphs and generalized wheel graphs.