On Friendly Index Sets of Broken Wheels with Three Spokes

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A \) be an abelian group. A labeling \( f: V(G) \to A \) induces an edge labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) + f(y) \) for each \( xy \in E(G) \). For each \( i \in A \), let \( v_f(i) = \text{card}\{v \in V(G) \mid f(v) = i\} \) and \( e_f(i) = \text{card}\{e \in E(G) \mid f^*(e) = i\} \). Let \( c(f) = \{\lvert e_f(i) – e_f(j) \rvert \mid (i, j) \in A \times A\} \). A labeling \( f \) of a graph \( G \) is said to be \( A \)-friendly if \( \lvert v_f(i) – v_f(j) \rvert \leq 1 \) for all \( (i, j) \in A \times A \). If \( c(f) \) is a \( (0, 1) \)-matrix for an \( A \)-friendly labeling \( f \), then \( f \) is said to be \( A \)-cordial. When \( A = \mathbb{Z}_2 \), the friendly index set of the graph \( G \), \( FI(G) \), is defined as \( \{\lvert e_f(0) – e_f(1) \rvert \mid \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly}\} \). In \([15]\) the friendly index set of a cycle is completely determined. We consider the friendly index sets of broken wheels with three spokes.