变换图G---的Wiener指标

Wiener Index of Transformation Graph G---

摘要: 图G的变换图G---的顶点集为V(G)∪E(G)，图G---中任意两顶点u,v∈V(G^---)只需满足下面任意一个条件便可以连边：1) u,v∈V(G)，它们在图G中不相邻，2)u,v∈E(G)，它们在图G中不相邻，3) u∈V(G)， v∈E(G)，它们在图G中不关联。图G的Wiener指标是图G中所有点对的距离之和。在本文中，我们确定了变换图G---是连通图时的Wiener指标。

Abstract: The transformation graph G--- of a graph G is the graph with vertex set V(G)∪E(G), in which two vertices u and v are joined by an edge if one of the following conditions holds: 1) u,v∈V(G) and they are not adjacent in G, 2) u,v∈E(G) and they are not adjacent in G, 3) one of u and v is in V(G) while the other is in E(G), and they are not incident in G. The Wiener index W(G) of G is the sum of the distances between all pairs of vertices in G. In this note, for any graph G, we de-termine the Wiener index of G---, when G--- is connected.