On \((1,2)\)-Strongly Indexable Spiders

For any integers \( k, d \geq 1 \), a \((p, q)\)-graph \( G \) with vertex set \( V(G) \) and edge set \( E(G) \), where \( p = |V(G)| \) and \( q = |E(G)| \), is said to be \((k, d)\)-strongly indexable (in short \((\textbf{k, d})\)-\textbf{SI}) if there exists a pair of functions \((f, f^+)\) that assigns integer labels to the vertices and edges, i.e., \( f: V(G) \to \{0, 1, \dots, p-1\} \) and \( f^+: E(G) \to \{k, k+d, k+2d, \dots, k+(q-1)d\} \), such that \( f^+(u, v) = f(u) + f(v) \) for any \((u, v) \in E(G)\). We determine here classes of spiders that are \((1, 2)\)-SI graphs. We show that every given \((1, 2)\)-SI spider can be extended to an \((1, 2)\)-SI spider with arbitrarily many legs.