Consider a simple graph G. We call a labeling w : E(G) ∪ V (G) → {1, 2, . . . , s} (total vertex) product-irregular, if all product degrees pdG(v) induced by this labeling are distinct, where pdG(v) = w(v) × Qe∋v w(e). The strength of w is s, the maximum number used to label the members of E(G) ∪ V (G). The minimum value of s that allows some irregular labeling is called the total vertex product irregularity strength and denoted tvps(G). We provide some general bounds, as well as exact values for chosen families of graphs.
