On friendly index sets of prisms and Mébius ladders

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 edge \( xy \in E(G) \). For \( i \in A \), let \( v_f(i) = |\{v \in V(G) : f(v) = i\}| \), and \( e_f(i) = |\{e \in E(G) : f^*(e) = i\}| \). Define \( c(f) = \{ |e_f(i) – e_f(j)| : (i, j) \in A \times A \} \).

A labeling \( f \) of a graph \( G \) is said to be **A-friendly** if \( |\text{v}_f(i) – \text{v}_f(j)| \leq 1 \) 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 \( G \), denoted \( FI(G) \), is defined as \( \{ |e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is } \mathbb{Z}_2\text{-friendly} \} \).

In this paper, we completely determine the friendly index sets for two classes of cubic graphs: prisms and Möbius ladders.