Linear Recurrent Sequences, Markov Chains and Their Applications in Graph Theory

TL;DR

The paper addresses sharpening the ergodic convergence of finite Markov chains by deriving a sub-exponential, spectral error term that depends on eigenvalue spacings, using an explicit Vandermonde inverse to obtain a spectral expression for recurrence coefficients.A complete treatment of linear recurrences is developed, including explicit formulas for the Vandermonde inverse and coefficients c_i, and the connection to powers of matrices via Cayley-Hamilton, enabling spectrally driven error bounds.These spectral tools are applied to Markov chains to produce a Convergence Theorem with explicit bounds in terms of the spectrum, and further extended to diameter bounds in directed graphs, recovering and extending known results for undirected graphs.The work highlights a deep link between recurrence theory, spectral graph theory, and network science, with potential practical impact on mixing times and graph diameter analyses in complex networks.

Abstract

After a brief review of the key theorems concerning recurrent sequences, we give an explicit computation of the inverse of the Vandermonde matrix. This will then be used to derive sub-exponential decay error terms in the ergodic theorem of Markov chains. Finally, we apply these results to give estimates for the diameters of directed graphs.

Theorems & Definitions (13)

• Lemma 1
• Proposition 2
• Theorem 3: Cayley-Hamilton
• Theorem 4
• Theorem 5: Fundamental Theorem of Markov Chains
• Theorem 6: Perron-Frobenius
• Remark 7
• Theorem 8: Convergence Theorem
• Remark 9
• Corollary 10