On $k,d$-Multiplicatively Indexable Graphs

Abstract

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.