On The Integer-magic Spectra of The Power of Paths

Abstract

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

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

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 $\mathbf{\text{integer-magic spectrum}}$ 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 $\text{k}\ge 4$ is proposed.