Integer-Magic Spectra of Functional Extensions of Graphs

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

$$ l^+(v) = \sum_{uv \in E(G)} l(uv) $$

is a constant map. For a given graph \( G \), the set of all \( k \in \mathds{N} \) for which \( G \) is \( \mathds{Z}_k \)-magic is called the integer-magic spectrum of \( G \) and is denoted by \( IM(G) \). In this paper, we will consider the functional extensions of \( P_n \) (\( n = 2, 3, 4 \)) and will determine their integer-magic spectra.