Integer-Magic Spectra of Functional Extensions of Graphs

Abstract

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

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

is a constant map. For a given graph $G$, the set of all $k\in \text{mathds}N$ for which $G$ is $\text{mathds}{Z}_k$-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.