On The Integer-Magic Spectra of Graphs

Abstract

For $k>0$, we call a graph $G=(V,E)$ as $\underset{\_}{Z_k-magic}$ if there exists a labeling $I:E(G)\to Z_k^{\ast }$ such that the induced vertex set labeling $I^{+}:V(G)\to Z_k$
$$I^{+}(v)=\mathrm{\Sigma }\{I(u,v):(u,v)\in E(G)\}$$
is a constant map. We denote the set of all $k$ such that $G$ is $k$-magic by $IM(G)$. We call this set as the integer-magic spectrum of $G$. We investigate these sets for general graphs.