The reflexive vertex strength on cycle and generalized friendship graph

Abstract

Let [Formula: see text] be a simple, un-directed and connected graph. The graph [Formula: see text] has a pair of sets [Formula: see text], where [Formula: see text] is nonempty vertex set and [Formula: see text] is an unordered pair of sets with two distinct vertices [Formula: see text] in [Formula: see text]. A total [Formula: see text]-labeling is defined as a function [Formula: see text] from the edge set to a set [Formula: see text] and a function [Formula: see text] from the vertex set to a set [Formula: see text], where [Formula: see text]. The total [Formula: see text]-labeling is a vertex irregular reflexive[Formula: see text]-labeling of the graph [Formula: see text], if for every two different vertices [Formula: see text] and [Formula: see text] of [Formula: see text], [Formula: see text], where [Formula: see text]. The reflexive vertex strength of the graph [Formula: see text], denoted by [Formula: see text] is the minimum [Formula: see text] for graph [Formula: see text] which has a vertex irregular reflexive [Formula: see text]-labeling. In this paper, we determined the exact value of the reflexive vertex strength of cycle and generalized friendship graph.