Vertex irregular reflexive labeling of disjoint union of gear and book graphs

Abstract

We define [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text]. The labeling [Formula: see text] is called a vertex irregular reflexive [Formula: see text]-labeling if any vertices have distinct weight which [Formula: see text] for any vertices in a graph [Formula: see text]. The weight of a vertex [Formula: see text] is defined as the sum of the labels of vertex and the labels of all edges incident this vertex. The smallest [Formula: see text] for which such labeling exists is called reflexive vertex strength of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the reflexive vertex strength of gear graphs, book graphs, triangular book graph and the disjoint union of gear graphs.