A proof of two conjectures on the Randić index and the average eccentricity

The Randić 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 in G and the summation extends over all edges u v of G . The eccentricity ϵ G ( v ) of a vertex v in G is the maximum distance from it to any other vertex, and the average eccentricity ϵ ̄ ( G ) in G is the mean value of the eccentricities of all vertices of G . There are two relations between the Randić index and the average eccentricity of connected graphs conjectured by a computer program called AGX: among the connected n -vertex graphs G , where n ≥ 3 , the maximum values of R ( G ) + ϵ ̄ ( G ) and R ( G ) ⋅ ϵ ̄ ( G ) are achieved only by a path. In this paper, we determine the graphs with the second largest average eccentricity and show that both conjectures are true.