Computing Edge-Weight Bounds of Antimagic Labeling on a Class of Trees

Jia-Bao Liu,Muhammad Javaid,Muhammad Kamran Aslam,Abdul Raheem

doi:10.1109/access.2019.2927244

Abstract

Graph labeling has wide applications in the field of computer science, such as coding theory, cryptography, software testing, database management systems, computer architecture, and networking. The computers connected in a network can now be converted in a graph and labels assigned to the graph so formed will help to regulate bandwidth, data traffic, in coding and decoding signals. Let A = (V(Λ), E(Λ)) beagraphwith|V(Λ)| = m and|V(Λ)| = n.Abijection from ζ : V(Λ)∪E(Λ) → {1, 2, 3, ⋯,m + n} is called (a, d)-edge antimagic total labeling if the edge-weights ζ(x) + ζ(xy) + ζ(y) for each xy ∈ E(Λ) form a sequence of consecutive positive integers with minimum edge-weight a and common difference d. In addition, it is called super (a, d)-edge antimagic total labeling if vertices receive the smallest labels. Enomoto et al. (2000) proposed the conjecture that every tree admits super (a, 0)-EAT labeling. In this note, bounds of the minimum and maximum edge-weights for super (a, d)-EAT labeling on the more generalized class of subdivided caterpillars are obtained. Moreover, we have investigated the existence of super (a, d)EAT labeling for the validation of the obtained bounds and the partial support of the aforesaid conjecture, where d ∈ {0, 1, 2}. In fact, the obtained results are a general extension of the results Akhlaq et al. [Utiltas Mathematica, 98 (2015), 227-249].

Highlights

In computer science, the role of graph labeling is improving the way in which repeated labels in data mining becomes negligible, security code to halt the attacks of hackers to precious data and in coding of data to transmit it to different networks and devices alike
Definition 2 [12]: An (a, d)-edge antimagic total labeling (a, d)-EATL of is defined as a bijective map ζ : V ( ) ∪
We have computed bounds for minimum and maximum edge-weights of S-(a, d)-EATL for more general class of subdivided caterpillar and proposed labeling scheme has validated the results of bounds

Summary

INTRODUCTION

The role of graph labeling is improving the way in which repeated labels in data mining becomes negligible, security code to halt the attacks of hackers to precious data and in coding of data to transmit it to different networks and devices alike. A mapping that maps non-zero positive integers onto the vertices, edges or both of a graph under certain conditions is called a labeling. It becomes total if we include both sets of vertices and edges as a domain. The concept of super (a, d)-EATL is a general extension of super EMTL given by Enomoto This historical back ground of the labeling of graphs includes the following conjectures for trees (connected and acyclic graphs): Conjecture 1. Akhlaq et al in [19] defined a subdivided caterpillar graph with the subdivision of the fixed number of pendant edges of a caterpillar, where a caterpillar is a tree with the peculiarity that the removal of its pendent vertices makes it a path.

PRELIMINARIES

MAIN RESULTS

CONCLUSION