On vertex integer-magic spectra of Caterpillar graphs

Abstract

Consider any graph \(G=(V(G), E(G))\) and \(k\) be any positive integer. Then a graph \(G\) is said to be \(\mathbb{Z}_k\)-vertex magic graph if there exist a map \(l: V(G) \longrightarrow \mathbb{Z}_k-\{0\}\) such that for any vertex \(v \in V(G)\), sum of the labels of vertices in the open neighborhood of \(v\) is a constant. ie, \(\omega(v)=\sum_{u \in N(v)} l(u)=\mu, \forall v \in V(G)\). The set \(\operatorname{VIM}(G)=\left\{k \in \mathbb{Z}^{+} \mid \mathrm{G}\right.\) is \(\mathbb{Z}_k\) - vertex magic \(\}\) is called vertex integer magic spectrum. In this paper, we determine VIM of caterpillar, super caterpillar and extended super caterpillar graphs.