On Multiplicative Labelings of a Graph

A \((p, q)\)-graph \( G \) is said to be multiplicative if its vertices can be assigned distinct positive integers so that the values of the edges, obtained as the products of the numbers assigned to their end vertices, are all distinct. Such an assignment is called a multiplicative labeling of \( G \). A multiplicative labeling is said to be \((a, r)\)-geometric if the values of the edges can be arranged as a geometric progression \( a, ar, ar^2, \dots, ar^{q-1} \). In this paper, we prove that some well-known classes of graphs are geometric for certain values of \( a, r \) and also initiate a study on the structure of finite \((a, r)\)-geometric graphs.