On \((a, d)\)- Antimagic Special Trees, Unicyclic Graphs and Complete Bipartite Graphs

Abstract

A connected graph \(G(V, E)\) is said to be \((a, d)\)-antimagic if there exist positive integers \(a\) and \(d\) and a bijection \(f: E \to \{1, 2, \ldots, |E|\}\) such that the induced mapping \(g_f: V \to \mathbb{N}\) defined by \(g_f(v) = \sum\{f(u,v) | (u, v) \in E(G)\}\) is injective and \(g_f(V) = \{a, a+d, a+2d, \ldots, a+(|V|-1)d\}\). In this paper, we mainly investigate \((a, d)\)-antimagic labeling of some special trees, complete bipartite graphs \(K_{m,n}\), and categorize \((a, d)\)-antimagic unicyclic graphs.