arXiv 1012.48562010-12-22math.CO

关于Randic指数与代数连通度关系的一个注记

Note on a relation between Randic index and algebraic connectivity

Xueliang Li, Yongtang Shi

备注：: 6 pages

摘要：

AutoGraphiX关于连通图$G$的Randić指数$R$与代数连通度$a$之间关系的一个猜想是：$$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\fracπ{n}})) $$具有相等性当且仅当$G$是$P_n$，这是Aouchiche和Hansen [M. Aouchiche和P. Hansen，谱图理论中的自动化算法综述，{ \it线性代数应用程序}。{ \bf 432}（2010），2293- 2322]。证明了该猜想对所有树和所有边连通度为$κ'(G)\geq 2$的连通图都成立，并且如果是$κ'(G)=1$，猜想对足够大的$n$也成立.该猜想也适用于所有具有直径$D\leq \frac {2(n-3+2\sqrt{2})}{π^2}$或最小度$δ\geq \frac n 2$的连通图。证明了$R\cdot a\geq \frac {8\sqrt{n-1}}{nD^2}$和$R\cdot a\geq \frac {nδ(2δ-n+2)} {2(n-1)}$，如果有$D\leq (n-1)^{1/4}$或$δ\geq \frac n 2-1$，则$R\cdot a$是最小的.

英文摘要：

A conjecture of AutoGraphiX on the relation between the Randić index $R$ and the algebraic connectivity $a$ of a connected graph $G$ is: $$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\fracπ{n}})) $$ with equality if and only if $G$ is $P_n$, which was proposed by Aouchiche and Hansen [M. Aouchiche and P. Hansen, A survey of automated conjectures in spectral graph theory, {\it Linear Algebra Appl.} {\bf 432}(2010), 2293--2322]. We prove that the conjecture holds for all trees and all connected graphs with edge connectivity $κ'(G)\geq 2$, and if $κ'(G)=1$, the conjecture holds for sufficiently large $n$. The conjecture also holds for all connected graphs with diameter $D\leq \frac {2(n-3+2\sqrt{2})}{π^2}$ or minimum degree $δ\geq \frac n 2$. We also prove $R\cdot a\geq \frac {8\sqrt{n-1}}{nD^2}$ and $R\cdot a\geq \frac {nδ(2δ-n+2)} {2(n-1)}$, and then $R\cdot a$ is minimum for the path if $D\leq (n-1)^{1/4}$ or $δ\geq \frac n 2-1$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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英文摘要：

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