The difference between remoteness and radius of a graph

Let GG be a connected graph of order n≥3n≥3. The remoteness ρ=ρ(G)ρ=ρ(G) is the maximum, over all vertices, of the average distance from a vertex to all others. The radius r=r(G)r=r(G) is the minimum, over all vertices, of the eccentricity of a vertex. Aouchiche and Hansen (2011) conjectured that ρ−r≥3−n4 if nn is odd and ρ−r≥2n−n24(n−1) if nn is even. In this paper, we confirm this conjecture. In addition, we completely characterize extremal graphs attaining the lower bound.