On Balance Index Sets of Generalized Book and Ear Expansion Graphs

Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces a partial edge labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(uv) = f(u) \) if and only if \( f(u) = f(v) \). For \( i \in \mathbb{Z}_2 \), let \( V_f(i) = \{v \in V(G) : f(v) = i\} \) and \( e_f(i) = |\{e \in E(G) : f^*(e) = i\}| \). A labeling \( f \) is called a friendly labeling if \( |V_f(0) – V_f(1)| \leq 1 \). The \( BI(G) \), the balance index set of \( G \), is defined as \( \{|e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly}\} \). This paper focuses on the balance index sets of generalized book and ear expansion graphs.