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 pdG(v) = ∏ e3v 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 str...

a vertex irregular total k-labeling of a graph g with vertex set v and edge set e is an assignment of positive integer labels {1, 2, ..., k} to both vertices and edges so that the weights calculated at vertices are distinct. the total vertex irregularity strength of g, denoted by tvs(g)is the minimum value of the largest label k over all such irregular assignment. in this paper, we study the to...

For a simple, undirected graph G with, at most one isolated vertex and no edges, labeling f:E(G)→{1,2,…,k1} of positive integers to the edges is called irregular if weights each has different value. The integer k1 then irregularity strength G. If number vertices in or order |G|, μ:E(G)→{1,2,…,k2} modular remainder divided by |G| k2 disjoint union two more graphs, denoted ‘+’, an operation where...