Proofs of conjectures on the Randic index and average eccentricity

The Randid index R(G) of a graph G is defined by Sigma uv 1/root d(u)d(v) (G) where d(u) is the degree of a vertex u and the summation extends over all edges uv of G. The eccentricity epsilon (v) of a vertex v is the maximum distance from it to any other vertex and the average eccentricity (epsilon) over bar (G) of graph G is the mean value of eccentricities of all vertices of G. There are relations between the Randic index and the average eccentricity of connected graphs conjectured by a computer program called AGX: for any connected graph G on n >= 14 vertices, both lower bounds of R(G) + (epsilon) over bar (G) and R(G) . (epsilon) over bar (G) are achieved only by a star. In this paper, we show that both conjectures are true. (C) 2015 Elsevier B.V. All rights reserved.