On Balance Index Sets of One-Point Unions of Graphs

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \), and let \( A = \{0, 1\} \). A labeling \( f: V(G) \to A \) induces an edge partial labeling \( f^*: E(G) \to A \) defined by \( f^*(xy) = f(x) \) if and only if \( f(x) = f(y) \) for each edge \( xy \in E(G) \). For each \( i \in A \), let

\[v_f(i) = |\{v \in V(G) : f(v) = i\}|\]

and

\[e_f(i) = |\{e \in E(G) : f^*(e) = i\}|.\]

The balance index set of \( G \), denoted \( BI(G) \), is defined as

\[\{|e_f(0) – e_f(1)|: |v_f(0) – v_f(1)| \leq 1\}.\]

In this paper, exact values of the balance index sets of five new families of one-point union of graphs are obtained, many of them, but not all, form arithmetic progressions.