Proofs of conjectures on the Randić index and average eccentricity

The Randic index R ( G ) of a graph G is defined by R ( G ) = ? u v 1 d ( u ) d ( v ) , where d ( u ) is the degree of a vertex u and the summation extends over all edges u v of G . The eccentricity ? ( v ) of a vertex v is the maximum distance from it to any other vertex and the average eccentricity ? ? ( 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 ) + ? ? ( G ) and R ( G ) ? ? ? ( G ) are achieved only by a star. In this paper, we show that both conjectures are true.