基于Bose-Mesner代数和Christoffel-Darboux恒等式的距离正则网络有效电阻的递推计算
Recursive calculation of effective resistances in distance-regular networks based on Bose-Mesner algebra and Christoffel-Darboux identity
摘要：
最近在\cite{jss1}中，作者给出了一种计算距离规则网络上的有效电阻（电阻距离）的方法，其中计算基于\cite{js}中引入的分层和与网络相关联的谱分布（Stieltjes函数）的Stieltjes变换。此外，在参考文献\cite{jss1}中，已经表明节点$α$和属于关于该$α$的相同层的所有节点$β$（$R_{αβ^{(i)}}$，属于关于$α$的第$i$层的$β$）之间的电阻距离是相同的。本文基于Bose-Mesner代数、网络分层、谱技术和Christoffel-Darboux恒等式，给出了一种递归计算任意距离正则电阻网络中电阻距离的算法。证明了距离正则网络的有效电阻是网络最短路距离的严格递增函数。换句话说，两点电阻$R_{αβ^{(m+1)}}$严格大于$R_{αβ^{(m)}}$。讨论了距离正则网络上的电阻距离与随机游动之间的联系，其中平均通勤时间（CT）及其平方根（称为欧氏通勤时间（ECT））作为距离与有效电阻有关.最后，对有限距离正则网络的一些重要例子，计算了电阻距离. { \bf关键词：阻力距离、关联模式、分层、距离-规则网络、Christoffel-Darboux恒等式} { \bf PACs Index：01.55.+ b，02.10.Yn}
英文摘要：
Recently in \cite{jss1}, the authors have given a method for calculation of the effective resistance (resistance distance) on distance-regular networks, where the calculation was based on stratification introduced in \cite{js} and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. Also, in Ref. \cite{jss1} it has been shown that the resistance distances between a node $α$ and all nodes $β$ belonging to the same stratum with respect to the $α$ ($R_{αβ^{(i)}}$, $β$ belonging to the $i$-th stratum with respect to the $α$) are the same. In this work, an algorithm for recursive calculation of the resistance distances in an arbitrary distance-regular resistor network is provided, where the derivation of the algorithm is based on the Bose-Mesner algebra, stratification of the network, spectral techniques and Christoffel-Darboux identity. It is shown that the effective resistance on a distance-regular network is an strictly increasing function of the shortest path distance defined on the network. In the other words, the two-point resistance $R_{αβ^{(m+1)}}$ is strictly larger than $R_{αβ^{(m)}}$. The link between the resistance distance and random walks on distance-regular networks is discussed, where the average commute time (CT) and its square root (called Euclidean commute time (ECT)) as a distance are related to the effective resistance. Finally, for some important examples of finite distance- regular networks, the resistance distances are calculated. {\bf Keywords: resistance distance, association scheme, stratification, distance-regular networks, Christoffel-Darboux identity} {\bf PACs Index: 01.55.+b, 02.10.Yn}
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