Expected hitting times for random walks on quadrilateral graphs and their applications

摘要

Let G be a connected graph. The quadrilateral graph of G, denoted by Q(G), is the graph obtained from G by replacing each edge in G with two parallel paths of lengths 1 and 3. In this paper, the complete information for the eigenvalues of the probability transition matrix of a random walk on Q(G) in terms of those of G is provided. Then the expected hitting time between any two vertices of Q(G) in terms of those of G is completely determined. Finally, as applications, the correlation between the degree-Kirchhoff index (resp. Kemeny's constant, number of spanning trees) of Q(G) and G is derived. Furthermore, based on the relationship of the expected hitting time between any two vertices of Q(G) and G, the resistance distance between any two vertices of Q(G) is presented in terms of that of G.