关于Randic指数及图的直径

On the Randic Index and Diameter of Graphs

设图G=(V,E)是简单图,其中V是顶点集,E是边集.对G中任意顶点v∈V,dv表示点v的度数.图G的Ran-dic指数也称为图G的连通性指数,定义为R=R(G)=∑uv∈E 1dndv.关于连通图的Randic指数R与直径D有如下猜想:R-D≥2-n2+1且DR≥21+n2--11,两个等式都成立当且仅当G≌Pn.本文将简化该猜想,并进一步证明当D≤2(n-1)23n-3+2 2或D≤n-3时。

Let G=（V,E） be a simple graph, where V is the vertex set, E is the edge set. The Randic index is defined as： R = R（G） R=R（G）=∑Nv∈E 1/√dndv A conjecture about the Randic index R and the diameter D of a connected graph is as follows：R-D≥√2-n＋1/2 and R/D≥1/2＋√2-1/n-1 , with equalities if and only if G is the path. In this paper, it is proved that this conjecture is true for all connected graphs withD≤[2（n-1）3/2/n-3＋2√2] or D≤n-3