A proof of two conjectures on the Randi膰 index and the average eccentricity

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• 作者：Meili Liang^a ; ^b ; ^{liangmeili2004@163.com} ; Bolian Liu^a ; ^{liubl@scnu.edu.cn}
• 关键词：Randi膰 index ; Conjecture ; Average eccentricity
• 刊名：Discrete Mathematics
• 出版年：2012
• 期刊代码：47_0012365x
• 类别：cp
• 出版时间：28 August, 2012
• 卷：312
• 期：16
• 页码：2446-2449
• 文件大小：195 K

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.