Group distance magic set of group vertex magic graphs

Let \(G\) be a graph of order \(n\) and let \(A\) be an additive Abelian group with identity 0. A mapping \(l : V(G) → A \{0\}\) is said to be a \(A\)-vertex magic labeling of \(G\) if there exists a \(μ ∈\) \(A\) such that \(w(v) = ∑u ∈ N_G(v) l(u) = μ\) for all \(v ∈ V\) and \(μ\) is called a magic constant of \(\). The group distance magic set of an \(A\)-vertex magic graph \(gdms(G,A)\) is defined as \(gdms(G,A):= \{ λ: λ is a magic constant of some A-vertex magic labeling \}\). In this paper, we investigate under what conditions \(gdms(G,A)\) is a subgroup of \(A\). We also introduce the concept of the reduced group distance magic set, \(rgdms(G, A)\), which can be used as a tool to determine \(gdms(G, A)\).