# Friendship-Like Graphs and It’s Classiffication

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 307)

## 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

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