Study of Random Walk Invariants for Spiro-Ring Network Based on Laplacian Matrices

Yasir Ahmad, Umar Ali, Daniele Ettore Otera () and Xiang-Feng Pan
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Yasir Ahmad: School of Mathematical Sciences, Anhui University, Hefei 230601, China
Umar Ali: Business School, University of Shanghai for Science and Technology, Shanghai 200093, China
Daniele Ettore Otera: Institute of Data Science and Digital Technologies, Vilnius University, 08412 Vilnius, Lithuania
Xiang-Feng Pan: School of Mathematical Sciences, Anhui University, Hefei 230601, China

Mathematics, 2024, vol. 12, issue 9, 1-19

Abstract: The use of the global mean first-passage time (GMFPT) in random walks on networks has been widely explored in the field of statistical physics, both in theory and practical applications. The GMFPT is the estimated interval of time needed to reach a state j in a system from a starting state i . In contrast, there exists an intrinsic measure for a stochastic process, known as Kemeny’s constant, which is independent of the initial state. In the literature, it has been used as a measure of network efficiency. This article deals with a graph-spectrum-based method for finding both the GMFPT and Kemeny’s constant of random walks on spiro-ring networks (that are organic compounds with a particular graph structure). Furthermore, we calculate the Laplacian matrix for some specific spiro-ring networks using the decomposition theorem of Laplacian polynomials. Moreover, using the coefficients and roots of the resulting matrices, we establish some formulae for both GMFPT and Kemeny’s constant in these spiro-ring networks.

Keywords: spiro-ring network; random walk; global mean first-passage time; Kemeny’s constant (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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