Cyclic Super Magic Labelings for Toroidal and Klein-Bottle Fullerenes

Qiang Pu,Muhammad Numan,Muhammad Asif,Saad Ihsan Butt,Fahad Rafique

doi:10.1109/access.2019.2943368

Abstract

A simple graph $G=(V,E)$ admits a cycle-covering if every edge in $E$ belongs at least to one subgraph of G isomorphic to a given cycle $C$ . The graph $G$ is $C$ -magic if there exists a total labeling $f:V \cup E \rightarrow \{1, 2, 3,\ldots, |V|+|E|\}$ such that for every subgraph $H'=(V', E')$ of $G$ isomorphic to $C$ , $\sum _{V\in V'}f(V)+\sum _{E\in E'}f(E)$ is constant, when $f(V)={1, 2, 3,\ldots, |V|}$ . Then $G$ is said to be $C$ -supermagic. In the present paper, we investigate the cyclic-supermagic behavior of toroidal and Klein-bottle graph.

Highlights

AND DEFINITIONS Fullerenes, the third type of carbon, have ended up essential atoms in science and innovation
We will use toroidal and Kelin bottle identification on Pmn that is first we identify the lateral sides to form cylinder and identify the top and bottom sides of the constructed cylinder in the same direction and in the opposite direction to form toroidal fullerenes Hmn and Klein-bottle fullerene Kmn respectively
At the end we use super cyclic magic graph definition and calculus to derive our results for toroidal and Klein-bottle fullerenes

Summary

Introduction

AND DEFINITIONS Fullerenes, the third type of carbon, have ended up essential atoms in science and innovation. In [1], [6], a massive literature is discussed about fullerenes along with their applications in science and innovation. Fullerenes are atoms made altogether out of carbon that were found in 1985 at Rice University. A large number of the viable uses of fullerenes take after straightforwardly from their uncommon properties. Significant chemical, physical and optical properties have been discussed in [7], which makes fullerenes key segments for the eventual fate of nano-electromechanical frameworks. The uses of fullerene are utilizations in solar cell, hydrogen gas storage devices, harden metals and alloys, interdigitated capacitors (IDCs), treatment of AIDS and in magnetic resonance imaging (MRI)

Methods

Conclusion