Product irregularity strength of certain graphs - Anholcer, Marcin (Ars mathematica contemporanea, 2014)
Consider a simple graph ?$G$? with no isolated edges and at most one isolated vertex. A labeling ?$w \colon E(G) → \{1, 2, …,m\}$? is called product - irregular, if all product degrees ?$pdG(v) = \prod_{e \ni v}w(e)$? 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 ?$ps(G)$?. We give the exact values of ?$ps(G)$? for several families of graphs, as complete bipartite graphs ?$K_{m, n}$?, where ?$2 \le m \le n \le {m + 2 \choose 2}$?, some families of forests, including complete ?$d$?-ary trees, and other graphs with ?$δ(G) = 1$?.
Keywords: graph theory; product irregularity strength; product-irregular labeling; teorija grafov; tree