Abstract
Among the various ideas that appear while studying graph theory, which has gained much attraction especially in graph labeling, labeling of graphs gives mathematical models which value for a vast range of applications in high technology (data security, cryptography, various problems of coding theory, astronomy, data security, telecommunication networks, etc.). A graph label is a designation of graph elements, i.e., the edges and/or vertex of a group of numbers (natural numbers), and is called assignment or labeling. The vertex or edge labeling is related to their domain asset of vertices or edges. Likewise, for total labeling, we take the domain as vertices and edges both at the same time. The reflexive edge irregularity strength (res) is total labeling in which weights of edges are not the same for all edges and the weight of an edge is taken as the sum of the edge labels and the vertices associated with that edge. In the res, the vertices are labeled with nonnegative even integers while the edges are labeled with positive integers. We have to make the labels minimum, whether they are associated with vertices or edges. If such labeling exists, then it is called the res of H and is represented as s res(H). In this paper, we have computed the res for the Cartesian product of path and cycle graph which is also known as generalizing prism.
Highlights
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
Ahmad et al defined on edge irregularity strength (es(H)) for any two edges u1u2 and u1′u2′ that the weights wφ(u1u2) and wφ(u1′u2′) are distinct, as weight for an edge u1u2 ∈ E(H) is wφ(u1u2) φ(u1) + φ(u2)
In [10], Baca 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 every edge has distinct weight
Summary
For total labeling, we take the domain as vertices and edges both at the same time. E reflexive edge irregularity strength (res) is total labeling in which weights of edges are not the same for all edges and the weight of an edge is taken as the sum of the edge labels and the vertices associated with that edge. The vertices are labeled with nonnegative even integers while the edges are labeled with positive integers. We have to make the labels minimum, whether they are associated with vertices or edges. If such labeling exists, it is called the res of H and is represented as s res(H). We have computed the res for the Cartesian product of path and cycle graph which is known as generalizing prism
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