On the vertex irregular reflexive labeling of several regular and regular-like graphs

Abstract

A total k-labeling is defined as a function g from the edge set to the first natural number ke and a function f from the vertex set to a non-negative even number up to 2kv , where k = max{ke , 2kv }. A vertex irregular reflexive k-labeling of the graph G is total k-labeling if wt(x) ¹ wt(x¢) for every two different vertices x and x¢ of G, where wt(x) = f (x) + Σ xy ∈E(G) g(xy). The reflexive vertex strength of the graph G, denoted by rvs(G), is the minimum k for a graph G with a vertex irregular reflexive k-labeling. We will determine the exact value of rvs(G) in this paper, where G is a regular and regular-like graph. A regular graph is a graph where each vertex has the same number of neighbors. A regular graph with all vertices of degree r is called an r-regular graph or regular graph of degree r. A regular-like graphs is an almost regular graph that we develop in a new definition and we called it with (s, r) -almost regular graphs.