【摘要】 随着有限图上随机游走理论的发展，作为衡量传输效率的关键参数平均首达时间成为了数学家们致力研究的重要课题。本文利用多项式方法研究图上的随机游走，推导出强正则图与完全图字典积对应的转移概率矩阵及其特征值的完整信息，进一步研究字典积图上任意两点间的平均首达时间及电阻距离的计算公式，得到了该字典积图的度积基尔霍夫指数、凯梅尼常数及该图的电阻直径。该代数方法得到的表达式大大简化了强正则图与完全图字典积上随机游走平均首达时间的计算。更多还原

【Abstract】 With the development of random walk theory on finite graphs, the first passage time, which is a key parameter to measure transmission efficiency, has become an important subject for mathematicians to study. The random walk on a graph is studied by mean of polynomial method. Transition probability matrix and complete information of its eigenvalues corresponding to the lexicographical product graph of the strongly regular graph and the complete graph are derived. Formulas for calculating the hitting time and the resistance distance between any two vertices on the lexicographical product graph are further studied. Furthermore, the Kirchhoff index, Kemeny constant and the resistance diameter of the lexicographic product graph are obtained. The expression obtained by this algebraic method greatly simplifies the calculation of the hitting time of the random walk on the lexicographical product of a strongly regular graph and a complete graph.更多还原