Integer-Magic Spectra of Amalgamations of Stars and Cycles

Abstract

For any positive integer \(k\), a graph \(G = (V, E)\) is said to be \(\mathbb{Z}_k\)-magic if there exists a labeling \(l: E(G) \to \mathbb{Z}_k – \{0\}\) such that the induced vertex set labeling \(l^+: V(G) \to \mathbb{Z}_k\) defined by

\[l^+(v) = \sum\{l(uv): uv \in E(G)\}\]

is a constant map. For a given graph \(G\), the set of all \(h \in \mathbb{Z_+}\) for which \(G\) is \(\mathbb{Z}_h\)-magic is called the integer-magic spectrum of \(G\) and is denoted by \(IM(G)\). In this paper, we will determine the integer-magic spectra of the graphs which are formed by the amalgamation of stars and cycles. In particular, we will provide examples of graphs that for a given \(n > 2\), they are not \(h\)-magic for all values of \(2 \leq k \leq n\).