摘要
The Randi膰 index of a graph is defined by , where is the degree of a vertex in and the summation extends over all edges of . The eccentricity of a vertex in is the maximum distance from it to any other vertex, and the average eccentricity in is the mean value of the eccentricities of all vertices of . There are two relations between the Randi膰 index and the average eccentricity of connected graphs conjectured by a computer program called AGX: among the connected -vertex graphs , where , the maximum values of and 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.