On Edge-Balance Index Sets of Flux Capacitors and \(L\)-products of Stars with Cycles

Let \( G \) be a simple graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( \mathbb{Z}_2 = \{0,1\} \). Any edge labeling \( f \) induces a partial vertex labeling \( f^+ : V(G) \to \mathbb{Z}_2 \) assigning \( 0 \) or \( 1 \) to \( f^+(v) \), \( v \) being an element of \( V(G) \), depending on whether there are more \( 0 \)-edges or \( 1 \)-edges incident with \( v \), and no label is given to \( f^+(v) \) otherwise. For each \( i \in \mathbb{Z}_2 \), let \( v_f(i) = |\{v \in V(G) : f^+(v) = i\}| \) and \( e_f(i) = |\{e \in E(G) : f(e) = i\}| \). An edge-labeling \( f \) of \( G \) is said to be edge-friendly if \( |e_f(0) – e_f(1)| \leq 1 \). The edge-balance index set of the graph \( G \) is defined as \( \text{EBI}(G) = \{\lvert v_f(0) – v_f(1) \rvert : f \text{ is edge-friendly}\} \). In this paper, we investigate and present results concerning the edge-balance index sets of flux capacitors and \( L \)-products of stars with cycles.