连通图的度距离和Wiener指数

Wiener index and degree distance of connected graphs

令P+(n)表示圈没有公共边的n阶连通图的集合,P+(n,m)表示P+(n)中具有m(m≥1)个极小圈的连通图集合.证明了当n≥6时,P+(n,m)中具有最小度距离的图是花F(n,m),它是m个具有一个公共顶点的三角形并在公共顶点粘上n-1-2m条悬挂边的图;同时证明P+(n)中具有最小度距离的图是F(n,1),它是一个三角形并在一个顶点上粘n-3条悬挂边的图.

Let P^＋ （n） be the set of conneted graphs whose each pair of minimal cycles have no common edges and P^＋ （ n, m） be the set of conneted graphs with m （ m ≥ 1 ） minimal cycles in P^＋ （n）. For n ≥6, we proved that the extremal graph with minimal degree distance in P^＋ （n, m） is a follower F（ n, m） which is m triangles sharing a common vertex on which n- 1 -2m pendent edges attached. Furthermore, we proved that the extremal graph with minimal degree distance in P ＋ （n） is the graph F（ n, 1 ） which consists of a triangle and n - 3 pendent edges attached to one vertex of it.