具有k个悬挂点的仙人掌图的调和指标

On the Harmonic Index of Cacti with k Pendent Vertices

图G的调和指标是指G所有边uv所对应的2/[d(u)+d(v)]之和,其中d(u),d(v)分别表示顶点u,v的度.一个连通的仙人掌图G是指它的任何两个圈至多只有一个公共顶点.主要采用归纳假设法,给出了具有k个悬挂点的所有仙人掌图的调和指标的极小值,并且刻画了相应达到其极小调和指标的极图.

The harmonic index of a graph G is defined as the sumof weights 2/[d（u）＋d（v）] of all edges uv of G, where d（u） and d （v） are the degrees of the vertices u and v in G, respectively. A connected graph G is a cactus if any two of its cycles have at most one common vertex. In this paper, we give the minimum value for the harmonic index of cacti of order n with k pendent vertices, and the corresponding extremal graphs are also characterized.