On Balance Index Sets of L-Products with Cycles and Complete Graphs

Let \( G \) be a graph with vertex set \( V \) and edge set \( E \). A labeling \( f : V \to \{0,1\} \) induces a partial edge labeling \( f^* : E \to \{0,1\} \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E \). The balance index set of \( G \), denoted \( \text{BI}(G) \), is defined as \( \{|f^{*-1}(0) – f^{*-1}(1)| : |f^{-1}(0) – f^{-1}(1)| \leq 1\} \). In this paper, we study the balance index sets of graphs which are \( L \)-products with cycles and complete graphs.