Kemeny's constant and the Kirchhoff index for the cluster of highly symmetric graphs

Kemeny's constant and Kirchhoffian indices for a family of non-regular graphs

We find closed form formulas for Kemeny's constant and its relationship with two Kirchhoffian indices for some composite graphs that use as basic building block a graph endowed with one of several symmetry properties.

Kemeny常数在马尔可夫链性质中的作用

The Role of Kemeny's Constant in Properties of Markov Chains

Kemeny和Snell（1960）首先证明了在具有平稳概率π_i和平均首次通过时间m_（ij）（当i = j时的平均递归时间）的有限状态不可约马尔可夫链中，\sum_j π_j m_（ij）是一个不依赖于i的常数K。这个常数后来被称为凯梅尼常数。各种技术找到表达式和各种界限K的派生。主要的解释集中在它的作用作为期望时间混合的马尔可夫链。各种应用被认为是包括扰动结果，混合的有向图和它的关系的基尔霍夫指数的正则图。

The expected values and limiting behaviours for the Gutman index, Schultz index, multiplicative degree-Kirchhoff index and additive degree-kirchhoff index of a random cyclooctane chain

In this paper, we first introduce the explicit analytical formulas for the expected values of the Gutman and Schultz indices for a random cyclooctane chain COCn. Meanwhile, the explicit formulas of the variances of the Gutman and Schultz indices for a random cyclooctane chain are determined and we prove these two indices are asymptotically subject to normal distribution. Furthermore, we are surprised to find the variances of Kf*(COCn) and Kf+(COCn) for a random cyclooctane chain based on the known results of others' paper and they are asymptotically subject to normal distribution.

马尔可夫链的树公式、平均首次通过时间和Kemeny常数

Tree formulas, mean first passage times and Kemeny's constant of a Markov chain

在本文中，我们的目标是提供概率和组合的见解树公式的格林函数和击中概率的马尔可夫链在有限的状态空间。这些树的公式是密切相关的环擦除随机游动的威尔逊算法的随机生成树，和混合时间的马尔可夫链树定理。令$m_{ij}$为具有有限状态空间$S$和转移矩阵$(p_{ij}; i, j \in S)$的不可约链从$i$到$j$的平均首次通过时间。众所周知，$m_{jj} = 1/π_j = Σ^{(1)}/Σ_j$，其中$π$是链的平稳分布，$Σ_j$是树积$\prod_{i \rightarrow k \in \textbf{t} }p_{ik}$和$Σ^{(1)}:= \sum_{j \in S} Σ_j$在$n^{n-2}$树$\textbf{t}$上的树和，$n^{n-2}$树$\textbf{t}$生成$S$，根$j$和边$i \rightarrow k$指向$j$。Chebotarev和Agaev从{ \em Kirchhoff矩阵树定理}导出了进一步的结果。我们推导出对于$i \ne j$，$m_{ij} = Σ_{ij}/Σ_j$，其中$Σ_{ij}$是与$Σ_j$相同的树积的$n^{n-2}$生成树的相同集合上的和，除了在每个乘积中因子$p_{kj}$被省略，其中$k = k(i,j,\textbf{t})$是$\textbf{t}$中从$i$到$j$的路径中$j$之前的最后一…