Geodesic
Total vertex product irregularity strength of graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1261-1276

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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 pd_G(v) induced by this labeling are distinct, where pd_G(v)=w(v)×∏_e∈ vw(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.
Keywords: product-irregular labeling, total vertex product irregularity strength, vertex-distinguishing labeling 
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Anholcer, Marcin; Emadi, Azam Sadat; Mojdeh, Doost Ali. Total vertex product irregularity strength of graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 4, pp. 1261-1276. http://geodesic.mathdoc.fr/item/DMGT_2024_44_4_a1/