On The Integer-magic Spectra of The Power of Paths

For \(\text{k}>0\), we call a graph G=(V,E) as \underline{\(\text{Z}_\text{k}\)-magic} if there exists an edge labeling \(\text{I: E(G)} \rightarrow \text{Z}_\text{k}^*\) such that the induced vertex set labeling \(\text{I}^+: \text{V(G)} \rightarrow \text{Z}_\text{k}\) defined by

\[\text{I}^+(\text{v}) = \Sigma \{(\text{I(u,v)) : (u,v)} \in \text{E(G)}\}\]

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 \textbf{integer-magic spectrum} of G. This paper deals with determining the integer-magic spectra of powers of paths \(\text{P}\text{n}^\text{k}\) for k=2 and 3. We also show that IM(\(\text{P}_{2\text{k}}^\text{k}) = \text{N}\setminus\{2\}\) for all odd integers \(\text{k}>1\). Finally, a conjecture for IM\((\text{P}_\text{n}^\text{k})\) for \(\text{k}\geq4\) is proposed.