Super (a; d)-star-antimagic graphs

Abstract

A simple graph $G=(V,E)$ admitting an $H$-covering is said to be $(a,d)$-$H$-antimagic if there exists a bijection $f:V\cup E\to\{1,2,...,|V|+|E|\}$ such that, for all subgraphs $H'$ of $G$ isomorphic to $H$, $wt_f(H')= \sum_{v\in V(H')} f(v) + \sum_{e\in E(H')} f(e)$, form an arithmetic progression $a,a + d,...,a+(t-1)d$, where $a$ is the first term, $d$ is the common difference and $t$ is the number of subgraphs in the $H$-covering. Then $f$ is called an $(a,d)$-$H$-antimagic labeling. If $f(V) = \{1, 2,..., |V|\}$, then $f$ is called super $(a,d)$-$H$-antimagic labeling.In this paper we investigate the existence of super (a,d)-star-antimagic labelings of a particular class of banana trees and construct a star-antimagic graph.