双圈图的极小广义Randi指标(英文)

Bicyclic Graphs with Minimum GeneralRandi Index

图G的广义R and i′c指标定义为Rα(G)=∑uv∈E(G)Rα(uv)=∑uv∈E(G)(d(u)d(v))α,其中d(u)是顶点u的度,α是实数.胡玉梅等给出了树的广义R and i′c指标的下界及其极图,吴宝音都仍等基本上给出了单圈图的广义R and i′c指标的下界及其极图.本文讨论双圈图G的R and i′c指标.利用吴宝音都仍的方法得到:当α>0时,Rα(G)≥6.6α+(n-5).4α(这里n=G).同时确定了这样的极图.

The general Randc index Ra（G） of a graph G is defined by Ra（G） =∑uv∈E（G）Ra（uv） =∑uv∈E（G） （d（u）d（v））^a where d（u） denotes the degree of a vertex u in G and a is an arbitrary real number. Hu, Li and Yuan gave the minimum general Randic index of trees and its extremal graph, and basically Wu and Zhang gave the minimum general Randic index of unicyclic graphs and its extremal graph. Let G be a bicyclic graph with the order n. In this paper, we use a method similar to that of Wu and Zhang to prove that Ra（G） ≥ 6 · 6^n ＋ （n - 5） · 4^n for a 〉 0, and give its extremal graphs.