On \(k,d\)-Multiplicatively Indexable Graphs

A \((p,q)\)-graph \( G \) is said to be \((k,d)\)-multiplicatively indexable if there exists an injection \( f : V(G) \to \mathbb{N} \) such that \( f^\times(E(G)) = \{k,k+d,\dots,k+(q-1)d\} \), where \( f^\times : E(G) \to \mathbb{N} \) is defined by \( f^\times(uv) = f(u)f(v) \) for every \( uv \in E(G) \). If further \( f(V(G)) = \{1,2,\dots,p\} \), then \( G \) is said to be a \((k,d)\)-strongly multiplicatively indexable graph. In this paper, we initiate a study of graphs that admit such labellings.