基于 Kirchhoff 指标的极值多边形链

Extremal Polygonal Chains with Respectto the Kirchhoff Index

摘要: 令 G 是一个简单连通图。 图 G 的 Kirchhoff 指标是图 G 中所有顶点对之间的电阻距离之和。 图 G 的电阻距离等效于将图 G 中的每条边替换为一个单位电阻后得到的电网络 N 中任意节点对之间的 有效电阻。 一个包含 n + 2 个多边形和 n + 1 个四边形的链，使得其中每个四边形的两条平行边各 与一个多边形有一条公共边，这样的链被称为多边形链。 本文利用电网络的标准技术和 S, T -同分 异构体的 Kirchhoff 指标的比较结果，刻画了 Kirchhoff 指标达到最大的极值多边形链为线性多 边形链 Ln， Kirchhoff 指标达到最小的极值多边形链为螺旋多边形链 Dn。 此结果推广了杨玉军 等人以及张雷雷刻画的基于 Kirchhoff 指标的极值亚苯基链的结果。

Abstract: Let G be a simple connected graph. The Kirchhoff index of a graph is the sum of the resistance distance between all vertex pairs in the graph G. The resistance distance in a graph is equivalent to the effective resistance between any node pairs in the electrical network obtained by replacing each edge in the graph G with a unit resistance. A chain containing n + 2 polygons and n + 1 squares such that each of the two parallel edges in squares has a common edge with a polygon is denoted as polygonal chains. In this paper, we use the standard techniques of electrical networks and the compare result of the Kirchhoff index of the S,T-isomers, characterize that the extremal polygonal chain with the maximal Kirchhoff index is the linear polygonal chain L_n and the extremal polygonal chain with the minimal Kirchhoff index is the helicene polygonal chain D_n, which generalizes the extremal Kirchhoff index of phenylene chains by Yujun Yang et al. and Leilei Zhang.