On Edge Irregular Reflexive Labeling for Generalized Prism

Alliance Manchester Business School, e University of Manchester, Manchester, UK Centre for Advanced Studies in Pure and Applied Mathematics, Bahauddin Zakariya University, Multan, Pakistan Department ofMathematics Education, Akenten Appiah-Menka University of Skills Training and Entrepreneurial Development, Kumasi 00233, Ghana Department of Mathematics, Comsats University Islamabad, Lahore Campus, Lahore, Pakistan


Introduction
Any graph H is the combination of vertices V(H) along with a possibly nonempty edge set E(H) of 2− element subsets of V(H). In this paper, all the chosen graphs are finite, without direction, nontrivial, connected, and simple (without loops and multiedges). For details about notations, see [1,2]. Nonnegative integers are used in this research. In 1988, Chartrand et al. [3] proposed the labeling problems in graph theory. Assign the edges positive integer to all connected simple graphs such as the graph became irregular. e irregular labeling is defined as ψ: E(H) ⟶ 1, 2, 3, . . . , m { } and is called irregular m− labeling for graph H if all the separate nodes u and u ′ have distinctly weights, that is, Lahel, in [4], studied, in detail, for the irregularity strength. For more results, see the works of Nierhoff in [5], Dimitz et al. in [6], Amar and Togni in [7], and Gyarfas in [8].
In [9], A. Ahmad et al. defined on edge irregularity strength (es(H)) for any two edges u 1 u 2 and u 1 ′ u 2 ′ that the weights w ϕ (u 1 u 2 ) and w ϕ (u 1 ′ u 2 ′ ) are distinct, as weight for an edge In [10], Bača et al. defined the parameter of total labeling for edge as well as vertex of graph and found the weights of an edge as sum of three integers which include the edge label and the labels of two vertices associated with that edge, and finally, every edge has distinct weight. For detailed studies on total edge irregularity strength, see [10,11]. e concept of total edge irregularity strength has been generalized by Zhang et al. in [12] for graph will be reflexive edge irregularity strength m− labeling.
If, for any graph H, the total m− labeling defined the mapping ψ e′ : E(H) ⟶ 1, 2, 3 . . . , m e′ and ψ v′ : V(H) ⟶ 0, 2, 4, . . . , 2m v′ , the mapping ψ is a total m− mapping of H such that ψ(a) � ψ v (a) if a ∈ V(H) and ψ(a) � ψ e (a) if a ∈ E(H), where k � max m e′ , 2m v′ . e total p− labeling ψ will be edge irregular reflexive p− labeling of the graph H if, for all the different edges say u 1 u 2 and u 1 ′ u 2 ′ , the weights w ϕ (u 1 u 2 ) and w ϕ (u 1 ′ u 2 ′ ) are not the same for every choice of edges where the weight for any edge suppose e smallest value of p for which such mapping exists is said to be res of the graph H and is represented by res(H). For details in reflexive edge irregularity strength, see [13][14][15][16][17].
In [12], the lemma is proven.

Lemma 1. For all graph say H,
In the present research paper, we have investigated the res for the Cartesian product of paths and cycles.

Definition.
e Cartesian product P and Q graphs is represented as P□Q and is the graph with vertices set V(P) × V(Q), with vertices (u 1 , u 1 ′ ) and (w 1 , w 1 ′ ) will be adjacent if and only if u 1 � w 1 and u 1 ′ w 1 ′ ∈ E(Q) or u 1 ′ � w 1 ′ and u 1 w 1 ∈ E(P).

Journal of Mathematics
Case 4. When d ≡ 1(mod 3), c is odd: An illustration of this reflexive labeling is shown in Figures 1 and 2.

Conclusion
In the present paper, we found the reflexive edge irregularity strength for generalized prism graph (P d □C c ), for d ≥ 3 and c ≥ 2.

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.