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arXiv 0906.5230math.CO
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Randić指数、直径和平均距离

Randić index, diameter and the average distance

Xueliang Li, Yongtang Shi

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7 pages
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摘要：

图$G$的Randić指数，记为$R(G)$，定义为$G$的所有边$uv$上的$1/\sqrt{d(u)d(v)}$之和，其中$d(u)$表示$G$中顶点$u$的度。本文部分地解决了关于图$G$的Randić指标$R(G)$与直径$D(G)$和平均距离$μ(G)$的关系的两个问题.本文证明了：对于任意阶为$n$且最小度为$δ(G)$的连通图$G$，若$δ(G)\geq 5$，则$R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$ ;若$δ(G)\geq n/5$和$n\geq 15$，则$\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$和$R(G)\geq μ(G)$ .此外，对于任意实数$\varepsilon \ (0<\varepsilon<1)$，如果$δ(G)\geq \varepsilon n$，则对于足够大的$n$，$\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$和$R(G)\geq μ(G)$成立。

英文摘要：

The Randić index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randić index $R(G)$ with relations to the diameter $D(G)$ and the average distance $μ(G)$ of a graph $G$. We prove that for any connected graph $G$ of order $n$ with minimum degree $δ(G)$, if $δ(G)\geq 5$, then $R(G)-D(G)\geq \sqrt 2-\frac{n+1} 2$; if $δ(G)\geq n/5$ and $n\geq 15$, $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq μ(G)$. Furthermore, for any arbitrary real number $\varepsilon \ (0<\varepsilon<1)$, if $δ(G)\geq \varepsilon n$, then $\frac{R(G)}{D(G)} \geq \frac{n-3+2\sqrt 2}{2n-2}$ and $R(G)\geq μ(G)$ hold for sufficiently large $n$.

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