On Friendly Index Sets of Root-unions of Stars By Cycles

Let \( G \) be a graph with vertex set \( V(G) \) and edge set \( E(G) \). A labeling \( f: V(G) \to \mathbb{Z}_2 \) induces an edge labeling \( f^*: E(G) \to \mathbb{Z}_2 \), defined by \( f^*(xy) = f(x) + f(y) \), for each edge \( xy \in E(G) \). For \( i \in \mathbb{Z}_2 \), let

\[
\text{v}_f(i) = \text{card}\{ v \in V(G) : f(v) = i \}
\]

and

\[
\text{e}_f(i) = \text{card}\{ e \in E(G) : f^*(e) = i \}.
\]

A labeling \( f \) of a graph \( G \) is said to be friendly if

\[
\lvert \text{v}_f(0) – \text{v}_f(1) \rvert \leq 1.
\]

The friendly index set of the graph \( G \), \( FI(G) \), is defined as

\[
\{ \lvert \text{e}_f(0) – \text{e}_f(1) \rvert : \text{the vertex labeling } f \text{ is friendly} \}.
\]

This is a generalization of graph cordiality. We introduce a graph construction called the root-union and investigate when gaps exist in the friendly index sets of root-unions of stars by cycles.