Arithmetic Labelings and Geometric Labelings of Finite Graphs

An injective map from the vertex set of a graph \( G \) to the set of all natural numbers is called an arithmetic/geometric labeling of \( G \) if the set of all numbers, each of which is the sum or product of the integers assigned to the ends of some edge, form an arithmetic/geometric progression. A graph is called arithmetic/geometric if it admits an arithmetic/geometric labeling. In this note, we show that the two notions just mentioned are equivalent—i.e., a graph is arithmetic if and only if it is geometric.