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On maximum indexable graphs

Let G=(V,E) be a graph with |V|=n and |E|=m. G is said to be maximum indexable if there exists a bijection f:V→{0,1,2,…,n−1} such that fmax:E→N is injective, where fmax(uv)=f(u)+f(v)+max{f(u),f(v)}. In this paper we prove that all trees and all unicyclic graphs are maximum indexable. We also construct several classes of maximum indexable graphs. We derive an explicit formula for λ(n), the maximum number of edges in a maximum indexable graph of order n.