On Edge Irregular Reflexive Labeling of Categorical Product of Two Paths
1 Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan
2 College of Computer Science and Information Technology, Jazan University, Jazan, Saudi Arabia
* Corresponding Author: Muhammad Ibrahim. Email: Received 19 October 2020; Accepted 13 November 2020; Issue published 18 January 2021
Abstract
Among the huge diversity of ideas that show up while studying graph theory, one that has obtained a lot of popularity is the concept of labelings of graphs. Graph labelings give valuable mathematical models for a wide scope of applications in high technologies (cryptography, astronomy, data security, various coding theory problems, communication networks, etc.). A labeling or a valuation of a graph is any mapping that sends a certain set of graph elements to a certain set of numbers subject to certain conditions. Graph labeling is a mapping of elements of the graph, i.e., vertex and/or edges to a set of numbers (usually positive integers), called labels. If the domain is the vertex-set or the edge-set, the labelings are called vertex labelings or edge labelings respectively. Similarly, if the domain is V (G)[E(G), then the labeling is called total labeling. A reflexive edge irregular k-labeling of graph introduced by Tanna et al.: A total labeling of graph such that for any two different edges ab and a'b' of the graph their weights has ωtχ(ab) = χ(a) + χ(ab) + χ(b) and ωtχ(a'b') = χ(a') + χ(a'b') + χ(b') are distinct. The smallest value of k for which such labeling exist is called the reflexive edge strength of the graph and is denoted by res(G). In this paper we have found the exact value of the reflexive edge irregularity strength of the categorical product of two paths Pa × Pb for any choice of a ≥ 3 and b ≥ 3.
Keywords
Edge irregular reflexive labeling; reflexive edge strength; categorical product of two paths 1 [click to view] This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.