Note on the group edge irregularity strength of graphs

Abstract

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