# Friendship-Like Graphs and It’s Classiffication

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## Abstract

A distance-compatible set-labeling (dcsl) of a connected graph *G* is an injective set assignment \(f : V(G) \rightarrow 2^{X},\) *X* being a nonempty set, such that the corresponding induced function \(f^{\oplus } :V(G)\times V(G) \rightarrow 2^{X}\) given by \(f^{\oplus }(uv)= f(u)\oplus f(v)\) satisfies \(\mid f^{\oplus }(uv) \mid = k_{(u,v)}^{f}d_{G}(u,v) \) for every pair of distinct vertices \(u, v \in V(G),\) where \(d_{G}(u,v)\) denotes the usual path distance between *u* and *v* and \(k_{(u,v)}^{f}\) is a constant. A dcsl *f* of *G* is *k*-uniform if all the constant of proportionality with respect to *f* are equal to *k*, and if *G* admits such a dcsl then *G* is called a *k*-uniform dcsl graph. And it has been already proved that, for a finite graph *G*, *k*-uniform dcsl graph *G* with is a finite set *X* if and only if *k*-embedding of *G* into a hypercube \({\mathbf {H}}(X)\). In this paper, we introduce Friendship-like graphs and its classification and prove that it admits 2-uniform dcsl.

## Keywords

*k*-Uniform dcsl graphs \(\lambda \)-Scale embeddable Friendship graphs

## Notes

### Acknowledgements

The authors are thankful to the Department of Science and Technology, Government of India, New Delhi, for the financial support concerning the Major Research Project (Ref: No. SR/S4/MS : 760/12).

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