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The Kemeny’s Constant and Spanning Trees of Hexagonal Ring Network

Shahid Zaman1, Ali N. A. Koam2, Ali Al Khabyah2, Ali Ahmad3,*
1 Department of Mathematics, University of Sialkot, Sialkot, 51310, Pakistan
2 Department of Mathematics, College of Science, Jazan University, New Campus, Saudi Arabia
3 College of Computer Sciences and Information Technology, Jazan University, Jazan, Saudi Arabia
* Corresponding Author: Ali Ahmad. Email:

Computers, Materials & Continua 2022, 73(3), 6347-6365.

Received 01 May 2022; Accepted 16 June 2022; Issue published 28 July 2022


Spanning tree () has an enormous application in computer science and chemistry to determine the geometric and dynamics analysis of compact polymers. In the field of medicines, it is helpful to recognize the epidemiology of hepatitis C virus (HCV) infection. On the other hand, Kemeny’s constant () is a beneficial quantifier characterizing the universal average activities of a Markov chain. This network invariant infers the expressions of the expected number of time-steps required to trace a randomly selected terminus state since a fixed beginning state . Levene and Loizou determined that the Kemeny’s constant can also be obtained through eigenvalues. Motivated by Levene and Loizou, we deduced the Kemeny’s constant and the number of spanning trees of hexagonal ring network by their normalized Laplacian eigenvalues and the coefficients of the characteristic polynomial. Based on the achieved results, entirely results are obtained for the Möbius hexagonal ring network.


Matrix Analysis; hexagonal ring network; Kemeny’s constant; Spanning tree

Cite This Article

S. Zaman, A. N. A. Koam, A. A. Khabyah and A. Ahmad, "The kemeny’s constant and spanning trees of hexagonal ring network," Computers, Materials & Continua, vol. 73, no.3, pp. 6347–6365, 2022.

This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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