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

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 [

All charts discussed in this paper are simple, finite and not directed. In Chartrand et al. [

There is an attraction arise from these papers as, an edge irregular

The total labeling of graph introduced by Bača et al. [

Keeping in view the problem imposed in Chartrand et al. [

Using both Chartrand et al. [

Thus, for a graph

In term of

where

Let us recall the following lemma proven in Ryan et al. [

From the above lemma it is noted that the minimum weight under

In this paper, we investigate the

As far as the category of graphs and full graph homomorphisms is concern, the categorical product was introduced by Culik et al. [

Next, we will show that

For this we define a

The weights of the edges are given as under:

Observe that the weights of all the links receive distinct values. So,

Next, we will show that

For this we define a

The weights of reflexive edges are given as under:

Observe that the weights of all the links receive distinct values. So,

Next, we will show that

For this we define a

The weights of reflexive edges are given as under:

Observe that the weights of all the links receive distinct values. So,

In this paper we have found the exact value of the reflexive edge irregularity strength of the categorical product of two paths