Super $(a,d)$ - Face Antimagic Total Labeling of Shackle of Cycle Graph

Abstract

A graph $G$ of order $p$, size $q$ and face $r$ is called a super $(a,d)$ - face antimagic total labelling, if there exist a bijection $f:V(G)\bigcup E(G)\bigcup F(G)$ $\rightarrow \{1,2,...,p+q+r\}$ such that the set of $s$-sided face weights,$W_{s}=\{a_{s},a_{s}+d,a_{s}+2d,...,a_{s}+(r_{s}-1)d\}$ form an arithmetic sequence for some integers as and common difference $d$ and $r_{s}$ is the number of $s$-sided faces. Such a graph is called super if the smallest possible labels appear on the vertices. In this paper we will study the existence on super $(a,d)$ - face antimagic total labeling of Shackle $C_6^1$ and it can be used to develop a secure poly alphabetic cryptosystem