On Friendly Index Sets of Generalized Books

Let \( G = (V, E) \) be a graph with a vertex labeling \( f: V \to \mathbb{Z}_2 \) that induces an edge labeling \( f^*: E \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) + f(y) \). For each \( i \in \mathbb{Z}_2 \), let

\[
v_f(i) = \text{card}\{v \in V: f(v) = i\}
\]

and

\[
e_f(i) = \text{card}\{e \in E: f^*(e) = i\}.
\]

A labeling \( f \) of a graph \( G \) is said to be friendly if

\[
\lvert v_f(0) – v_f(1) \rvert \leq 1.
\]

The friendly index set of \( G \) is defined as

\[
\{\lvert e_f(1) – e_f(0) \rvert : \text{the vertex labeling } f \text{ is friendly}\}.
\]

In this paper, we determine the friendly index sets of generalized books.