Full Friendly Index Set-II

Let \( G = (V, E) \) be a graph. A vertex labeling \( f: V \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E \). For each \( i \in \mathbb{Z}_2 \), define \( v_f(i) = |f^{-1}(i)| \) and \( e_f(i) = |{f^*}^{-1}(i)| \). We call \( f \) friendly if \( |v_f(1) – v_f(0)| \leq 1 \). The full friendly index set of \( G \) is the set of all possible values of \( e_f(1) – e_f(0) \), where \( f \) is a friendly labeling. In this paper, we study the full friendly index set of the wheel \( W_n \), the tensor product of paths \( P_2 \) and \( P_n \), i.e., \( P_2 \otimes P_n \), and the double star \( D(m, n) \).