Group distance magic set of group vertex magic graphs

<p>Let <span class="math inline">\(G\)</span> be a graph of order <span class="math inline">\(n\)</span> and let <span class="math inline">\(A\)</span> be an additive Abelian group with identity 0. A mapping <span class="math inline">\(l : V(G) \to A \setminus \{0\}\)</span> is said to be a <span class="math inline">\(A\)</span>-vertex magic labeling of <span class="math inline">\(G\)</span> if there exists a <span class="math inline">\(\mu \in\)</span> <span class="math inline">\(A\)</span> such that <span class="math inline">\(w(v) = \sum\limits_{u \in N_G(v)} l(u) = \mu\)</span> for all <span class="math inline">\(v \in V\)</span> and <span class="math inline">\(\mu\)</span> is called a magic constant of <span class="math inline">\(\ell\)</span>. The group distance magic set of an <span class="math inline">\(A\)</span>-vertex magic graph <span class="math inline">\(gdms(G,A)\)</span> is defined as <span class="math inline">\(gdms(G,A):= \{ \lambda: \lambda \text{ is a magic constant of some $A$-vertex magic labeling} \}\)</span>. In this paper, we investigate under what conditions <span class="math inline">\(gdms(G,A)\)</span> is a subgroup of <span class="math inline">\(A\)</span>. We also introduce the concept of the reduced group distance magic set, <span class="math inline">\(rgdms(G, A)\)</span>, which can be used as a tool to determine <span class="math inline">\(gdms(G, A)\)</span>.</p>