On Edge Irregular Reflexive Labeling of Categorical Product of Two Paths Muhammad Javed Azhar Khan 1 , Muhammad Ibrahim 1,* and Ali Ahmad 2 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: mibtufail@gmail.com Received: 19 October 2020; Accepted: 13 November 2020 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 ele- ments 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 label- ings 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 wt 1 ab ð Þ¼ 1a ð Þþ 1 ab ð Þþ 1b ðÞ and wt 1 a 0 b 0 ð Þ¼ 1a 0 ð Þþ 1a 0 b 0 ð Þþ 1b 0 ð Þ are distinct. The smallest value of k for which such labeling exist is called the reflex- ive 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 P a  P b ð Þ for any choice of a  3 and b  3: Keywords: Edge irregular reflexive labeling; reflexive edge strength; categorical product of two paths 1 Introduction The area of graph theory has experienced fast developments during the last 60 years. Among the huge diversity of concepts that appear while studying this subject one that has gained a lot of popularity is the concept of labeling of graphs with more than 1700 papers in the literature and a very complete dynamic survey by Gallian [1]. This new branch of mathematics has caught the attention of many authors and many new labeling results appear every year. This popularity is not only due to the mathematical challenges of graph labeling, but also for the wide range of its application, for instance X-ray crystallography, coding theory, radar, astronomy, circuit design, network design and communication design. In fact, Bloom et al. studied applications of graph labeling to other branches of science and it is possible to find part of this work in Bloom et al. [2] and Bloom et al. [3]. 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. Computer Systems Science & Engineering DOI:10.32604/csse.2021.014810 Article ech T Press Science