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

ABSTRACT

Let G be a connected graph. The quadrilateral graph of G, denoted by 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 in terms of those of G is provided. Then the expected hitting time between any two vertices of 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 and G is derived. Furthermore, based on the relationship of the expected hitting time between any two vertices of and G, the resistance distance between any two vertices of is presented in terms of that of G.

KEYWORDS:

• Hitting time
• quadrilateral graph
• degree-Kirchhoff index
• Kemeny’s constant
• spanning tree

AMS SUBJECT CLASSIFICATIONS:

• 05C50
• 60J20

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

S.L. acknowledge the financial support from the National Natural Science Foundation of China [grant number 11671164], [grant number 11271149] and the Program for New Century Excellent Talents in University [grant number NCET-13-0817].