Proofs of conjectures on the Randić index and average eccentricity

The Randić index R(G)R(G) of a graph GG is defined by R(G)=∑uv1d(u)d(v), where d(u)d(u) is the degree of a vertex uu and the summation extends over all edges uvuv of GG. The eccentricity ϵ(v)ϵ(v) of a vertex vv is the maximum distance from it to any other vertex and the average eccentricity ϵ̄(G) of graph GG is the mean value of eccentricities of all vertices of GG. There are relations between the Randić index and the average eccentricity of connected graphs conjectured by a computer program called AGX: for any connected graph GG on n≥14n≥14 vertices, both lower bounds of R(G)+ϵ̄(G) and R(G)⋅ϵ̄(G) are achieved only by a star. In this paper, we show that both conjectures are true.