Product irregularity strength of graphs

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w : E ( G ) → { 1 , 2 , … , m } is called product-irregular, if all product degrees p d G ( v ) = ∏ e ∋ v w ( e ) are distinct. The goal is to obtain a product-irregular labeling that minimizes the maximum label. This minimum value is called the product irregularity strength. The analogous concept of irregularity strength, with sums in place of products, has been introduced by Chartrand et al. and investigated by many authors.