图的Wiener指标和Kirchhoff指标

发布时间：2018-02-12 05:45

  本文关键词： 电阻距离 Wiener指标 Krichhoff指标 单圈图 Nordhaus-Gaddum型结果　出处：《烟台大学》2017年硕士论文　论文类型：学位论文

【摘要】：设G是一个连通图,其顶点集合为V(G).对G中任意两个顶点i和j,i和j之间的距离定义为连接这两个顶点之间的最短路的长度,而i和j之间得电阻距离定义为用单位电阻代替G中的每条边后所得的电网络中这两个节点之间的等效电阻.图G的Wiener指标,记作W(G),定义为G中所有顶点之间的距离的和.图G的Kirchhoff指标,记作Kf(G),定义为G中所有顶点之间的电阻距离之和.图的Wiener指标和Kirchhoff指标是图的重要不变量,并且在化学里QSAR和QSPR的研究中具有广泛的应用.本文主要研究图的Wiener指标和Kirchhoff指标,具体研究内容如下所述.首先,我们刻画具有第四大和第四小Wiener指标的单圈图.只含有一个圈的图称作单圈图.在之前的研究中,具有前三大和前三小Wiener指标的单圈图已经刻画清楚.沿着这个方向,我们进一步刻画具有第四大和第四小Wiener指标的单圈图.本文证明了在所有顶点数≥8的单圈图中,C_5(S_(n-4)和C_2~(u1,u2)(S_3,S_(n-4))具有第四小的Wiener指标,而C_3~(u1,u2)(P_3,P_(n-4))具有第四大的Wiener指标.其次,我们完全解决了关于Kirchhoff指标的Nordhaus-Gaddum型结果的一个猜想.在2011年,Yang,Zhang和Klein[Y.Yang,H.Zhang,D.J.Klein,New Nordhaus-Gaddum-type results for the Kirchhoff index,J.Math Chem.49(2011)1587-1598]提出了关于Kirchhoff指标的NordhausGaddum型结果的一个猜想.他们猜想一个图G及其补图(?)的Kirchhoff指标的和达到最大当且仅当G是路P_n或者其补图(?).在文本中,应用图论和电网络的方法和技巧,我们完全解决了该猜想.
[Abstract]:Let G be a connected graph, and the vertex set of G is VG. The distance between any two vertices I, J and j in G is defined as the length of the shortest path between the two vertices. And the resistance distance between I and j is defined as the equivalent resistance between the two nodes in the electric network after replacing each edge of G with the unit resistor. The Wiener index of figure G. The Kirchhoff index of a graph G is defined as the sum of the distances between all vertices in G. it is defined as the sum of resistance distances between all vertices in G. the Wiener index and the Kirchhoff index of a graph are important invariants of a graph. And it is widely used in the study of QSAR and QSPR in chemistry. This paper mainly studies the Wiener index and Kirchhoff index of the graph, the specific research contents are as follows. We characterize unicyclic graphs with 4th and 4th small Wiener indices. Graphs with only one cycle are called unicyclic graphs. In previous studies, unicyclic graphs with the first three largest and the first three small Wiener indices have been clearly characterized. We further characterize unicyclic graphs with 4th large and 4th small Wiener indexes. In this paper, we prove that in all unicycle graphs with vertex number 鈮，

本文编号：1504915