Note on the group edge irregularity strength of graphs

TL;DR

This paper explores the edge group irregularity strength of graphs, providing bounds on this parameter and related graph invariants, which helps in understanding labelings that distinguish edges via group sums.

Contribution

It introduces new upper bounds for the edge group irregularity strength and related parameters, advancing the theoretical understanding of graph labelings with group elements.

Findings

Derived upper bounds for $es_g(G)$ and $es(G)$

Established bounds for the harmonious order $ m{har}(G)$

Contributed to the theory of graph labelings with Abelian groups

Abstract

We investigate the \textit{edge group irregularity strength} ($es_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\mathcal{G}$ of order $s$, there exists a function $f:V(G)\rightarrow \mathcal{G}$ such that the sums of vertex labels at every edge are distinct. In this note we provide some upper bounds on $es_g(G)$ as well as for edge irregularity strength $es(G)$ and harmonious order $\rm{har}(G)$.