On (a; d)-eat labeling of subdivision of trees

Abstract

An (a, d)-edge antimagic total (EAT) labeling on a graph ? with p vertices and q edges is a one-to-one function ? from V (?)?E(?) onto the set of integers 1, 2, ...p+q with the property that for each edge uv, the set {?(u) +?(uv) +?(v) : uv ? E(?)} form an arithmetic progression (A. P.) starting with a and having common difference d, where a > 0 and d ? 0 fixed integers. A (a, d)-EAT labeling is called super (a, d)-EAT labeling if the smallest numbers are labels to the vertices. In this paper, we have to show that the graph of the subdivided star and subdivided caterpillar are super (a, d)-EAT labeling.