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 k(e) and a function f from the vertex set to a non-negative even number up to 2k(v), where k = max{k(e), 2k(v)}. A vertex irregular reflexive k-labeling of the graph G is total k-labeling if wt(x) not equal wt(x') for every two different vertices x and x' of G, where wt(x) = f(x) + Sigma(xy is an element of 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.