Product irregularity strength of certain graphs
Marcin Anholcer · Ars Mathematica Contemporanea · 2013
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 maximal label. This minimal value is called the product irregularity strength and denoted p s ( G ) . We give the exact values of p s ( G ) for several families of graphs, as complete bipartite graphs K m , n , where 2 ≤ m ≤ n ≤ ( m + 2) choose 2, some families of forests, including complete d -ary trees, and other graphs with δ ( G ) = 1 .