On Balance Index Sets of Generalized Wheels

A vertex labeling \( f: V \to \{0,1\} \) of the simple graph \( G = (V, E) \) induces a partial edge labeling \( f^*: E \to \{0,1\} \) defined by \( f^*(uv) = f(u) \) if and only if \( f(u) = f(v) \). Let \( v(i) \) and \( e(i) \) be the number of vertices and edges, respectively, that are labeled \( i \), and define the balance index set of \( G \) as \( \{|e(0) – e(1)| : |v(0) – v(1)| \leq 1\} \). In this paper, we determine the balance index sets of generalized wheels, which are the Zykov sum of a cycle with a null graph.