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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 a partial edge labeling \( f^*: E(G) \to A \) defined by
\[
f^*(xy) = f(x), \text{ if and only if } f(x) = f(y),
\]
for each edge \( xy \in E(G) \). For \( i \in A \), let
\[
v_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}
\]
and
\[
e_f^*(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.
\]
A labeling \( f \) of a graph \( G \) is said to be friendly if
\[
|v_f(0) – v_f(1)| \leq 1.
\]
If
\[
|e_f(0) – e_f(1)| \leq 1,
\]
then \( G \) is said to be \textbf{\emph{balanced}}. The \textbf{\emph{balance index set}} of the graph \( G \), \( BI(G) \), is defined as
\[
BI(G) = \{ |e_f(0) – e_f(1)| : \text{the vertex labeling } f \text{ is friendly} \}.
\]
Results parallel to the concept of friendly index sets are pr