On Zero Magic Sums of Integer Magic Graphs

Abstract

For a positive integer \(k\), let \(\mathbb{Z}_k = (\mathbb{Z}_k, +, 0)\) be the additive group of congruences modulo \(k\) with identity \(0\), and \(\mathbb{Z}_1\) is the usual group of integers \(\mathbb{Z}\). We call a finite simple graph \(G = (V(G), E(G))\) \(\mathbb{Z}_k\)-magic if it admits an edge labeling \(\ell: E(G) \to \mathbb{Z}_k \setminus \{0\}\) such that the induced vertex sum labeling \(\ell^+: V(G) \to \mathbb{Z}_k\), defined by \(\ell^+(v) = \sum_{uv \in E(G)} \ell(uv)\), is constant. The constant is called a \emph{magic sum index}, or an \emph{index} for short, of \(G\) under \(\ell\), following R. Stanley. The \emph{null set} of \(G\), defined by E. Salehi as the set of all \(k\) such that \(G\) is \(\mathbb{Z}_k\)-magic with zero magic sum index, is denoted by \(Null(G)\). For a fixed integer \(k\), we consider the set of all possible magic sum indices \(r\) such that \(G\) is \(\mathbb{Z}_k\)-magic with magic sum index \(r\), and denote it by \(I_k(G)\). We call \(I_k(G)\) the \emph{index set} of \(G\) with respect to \(\mathbb{Z}_k\). In this paper, we investigate properties and relations of index sets \(I_k(G)\) and null sets \(Null(G)\) for \(\mathbb{Z}_k\)-magic graphs. Among others, we determine null sets of generalized wheels and generalized fans and construct infinitely many examples of \(\mathbb{Z}_k\)-magic graphs with magic sum zero. Some open problems are presented.