Anti-Magic Labeling on Harary Graph and Subdivided Caterpille

Abstract

Graphs play an important and vital role in Mathematics. Graph labeling creates new ideas graph theory. Can labeling be helped us to solve problems in Mathematics and other fields? The answer of this question increases the deep value of Graph Labeling. A magic labeling of a graph is described by Kotzig and Rosa [22] in 1970 as an important iniative. A graph labeling (or valuation) is a unique way that gives the labels to graph elements (commonly, these are non negative or positive integers). if we consider the domain set of all vertices and edges; such labelings are called total labelings. Sometime we use labelings for vertex set or the edge set alone and these labelings would be called as vertex-labeling and edge-labeling respectively. Many other domains are also possible. There are some famous labelings like as harmonious, cordial, graceful and antimagic. In 2000, Baˇca et al. [2] discussed a (a, d)-vertex-antimagic total labeling. A (a, d)- vertex-antimagic total labeling is called super (a, d)-vertex-antimagic total labeling, when we give smallest labels to the vertices. An (a, d)-edge-antimagic total labeling was defined by Simanjuntak et al. [41]. An (a, d)-edge-antimagic total labeling is called a super (a, d)-edge-antimagic total labeling, when we give the least labels to the vertices. A graph G, which is made up of cycles with chords is known as Harary Graph. A brief discussion exposed important results on super (a, d)-edge-antimagic total labeling Harary graph by M. Baˇca and M. Murugan [1, 6]. This work is designed for the construction of (a, d)-vertex-antimagic total labeling of Harary graph as well as for super (a, d)-vertex antimagic total labeling of Harary graph for the different value of d.