On Super \( a, d \)-edge Antimagic Total Labeling of Disconnected Graphs

Abstract

A graph \( G \) is called \((a,d)\)-\text{edge antimagic total} \((a, d)\)-\text{EAT} if there exist integers \( a > 0, d \geq 0 \) and a bijection \( \lambda: V \cup E \rightarrow \{1,2,\ldots,|V| + |E|\} \) such that

\[
W = \{w(xy) : xy \in E\} = \{a,a+d,\ldots,a+(|E|-1)d\},
\]

where \( w(xy) = \lambda(x) + \lambda(y) + \lambda(xy) \). An \((a, d)\)-EAT labeling \( \lambda \) of graph \( G \) is \text{super} if \( \lambda(V) = \{1,2,\ldots,|V|\} \). In this paper, we describe how to construct a super \((a, d)\)-EAT labeling on some classes of disconnected graphs, namely \( P_n \cup P_{n+1} \), \( nP_2 \cup P_n \), and \( nP_2 \cup P_{n+2} \), for positive integer \( n \).