NoteA 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)=∑uv1d(u)d(v), where d(u) is the degree of a vertex u in G and the summation extends over all edges uv of G. The eccentricity ϵG(v) of a vertex v in G is the maximum distance from it to any other vertex, and the average eccentricity ϵ̄(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)+ϵ̄(G) and R(G)⋅ϵ̄(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.