Total vertex product irregularity strength of graphs

TL;DR

This paper introduces the concept of total vertex product irregularity strength in graphs, establishing bounds and exact values for specific graph families to distinguish vertices via product-based labelings.

Contribution

It defines the total vertex product irregularity strength and provides bounds and exact values for various families of graphs, advancing vertex-distinguishing labeling theory.

Findings

Established bounds for total vertex product irregularity strength.

Calculated exact values for specific graph families.

Enhanced understanding of product-based vertex distinguishing labelings.

Abstract

Consider a simple graph $G$. We call a labeling $w:E(G)\cup V(G)\rightarrow \{1, 2, \dots, s\}$ (\textit{total vertex}) \textit{product-irregular}, if all product degrees $pd_G(v)$ induced by this labeling are distinct, where $pd_G(v)=w(v)\times\prod_{e\ni v}w(e)$. The strength of $w$ is $s$, the maximum number used to label the members of $E(G)\cup V(G)$. The minimum value of $s$ that allows some irregular labeling is called \textit{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.