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.