Proofs of conjectures on the Randić index and average eccentricity

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• 作者：Meili Liang ; Jianxi Liu ; ^{liujianxi2001@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Conjecture ; Randić index ; Average eccentricity
• 刊名：Discrete Applied Mathematics
• 出版年：2016
• 出版时间：31 March 2016
• 年：2016
• 卷：202
• 期：Complete
• 页码：188-193
• 全文大小：354 K

文摘

The Randić index R(G)$R\left(G\right)$ of a graph G$G$ is defined by $R\left(G\right)={\sum }_{uv}\frac{1}{\sqrt{d\left(u\right)d\left(v\right)}}$, where d(u)$d\left(u\right)$ is the degree of a vertex fbcb" title="Click to view the MathML source">u$u$ and the summation extends over all edges uv$uv$ of G$G$. The eccentricity ϵ(v)$ϵ\left(v\right)$ of a vertex v$v$ is the maximum distance from it to any other vertex and the average eccentricity $\overline{ϵ}\left(G\right)$ of graph G$G$ is the mean value of eccentricities of all vertices of G$G$. 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 G$G$ on n≥14$n\ge 14$ vertices, both lower bounds of $R\left(G\right)+\overline{ϵ}\left(G\right)$ and $R\left(G\right)\cdot \overline{ϵ}\left(G\right)$ are achieved only by a star. In this paper, we show that both conjectures are true.