# Δ-Optimum Exclusive Sum Labeling of Certain Graphs with Radius One

## Abstract

A mapping *L* is called a sum labeling of a graph *H*(*V*(*H*),*E*(*H*)) if it is an injection from *V*(*H*) to a set of positive integers, such that *xy* ∈ *E*(*H*) if and only if there exists a vertex *w* ∈ *V*(*H*) such that *L*(*w*) = *L*(*x*) + *L*(*y*). In this case, *w* is called a *working vertex*. We define *L* as *an exclusive sum labeling* of a graph *G* if it is a sum labeling of \(G\cup\bar K_{r}\) for some non negative integer *r*, and *G* contains no working vertex. In general, a graph *G* will require some isolated vertices to be labeled exclusively. The least possible number of such isolated vertices is called *exclusive sum number* of *G*; denoted by *ε*(*G*).

An exclusive sum labeling of a graph *G* is said to be *optimum* if it labels *G* exclusively by using *ε*(*G*) isolated vertices. In case *ε* (*G*) = Δ (*G*), where Δ(*G*) denotes the maximum degree of vertices in *G*, the labeling is called Δ*-optimum exclusive sum labeling*.

In this paper we present Δ-optimum exclusive sum labeling of certain graphs with radius one, that is, graphs which can be obtained by joining all vertices of an integral sum graph to another vertex. This class of graphs contains infinetely many graphs including some populer graphs such as wheels, fans, friendship graphs, generalised friendship graphs and multicone graphs.

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