Study On Proximity And Average Eccentircity Of A Graph

Let G=(V, E) be a connected graph on n vertices. The proximity π(G) of Gis the minimum average distance from a vertex of G to all others. The eccentricityeccG(v) of a vertex v in G is the largest distance from v to another vertex, and theaverage eccentricity ecc(G) of the graph G is1∑nv∈V (G)eccG(v). Recently, it wasconjectured by Aouchiche and Hansen [1] that for any connected graph G on n≥3vertices, ecc(G) π(G)≤ecc(P_n) π(_Pn), with equality if and only if G～=Pn. Inthis paper, we show that this conjecture is true.