On Balance Index Sets of Trees of Diameter Four

Let \( G \) be a simple graph with a vertex set \( V(G) \) and an edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). A labeling \( f : V(G) \to \mathbb{Z}_2 \) induces an edge partial labeling \( f^* : E(G) \to \mathbb{Z}_2 \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = \lvert \{v \in V(G) : f(v) = i\} \rvert \) and \( e_f(i) = \lvert \{e \in E(G) : f^*(e) = i\} \rvert \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{\lvert e_f(0) – e_f(1) \rvert : \lvert v_f(0) – v_f(1) \rvert \leq 1\} \). In this paper, we investigate and present results concerning the balance index sets of trees of diameter four.