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Spectral convergence of random regular graphs: Chebyshev polynomials, non‐backtracking walks, and unitary‐color extensions

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  • Yulin Gong
  • Wenbo Li
  • Shiping Liu

Abstract

In this paper, we extend a criterion of Sodin on the convergence of graph spectral measures to regular graphs of growing degree. As a result, we show that for a sequence of random (qn+1)$(q_n+1)$‐regular graphs Gn$G_n$ with n$n$ vertices, if qn=no(1)$q_n = n^{o(1)}$ and qn$q_n$ tends to infinity, the normalized spectral measure converges almost surely in p$p$‐Wasserstein distance to the semicircle distribution for any p∈[1,∞)$p \in [1, \infty)$. This strengthens a result of Dumitriu and Pal. Many of the results are also extended to unitary‐colored regular graphs. For example, we give a short proof of the weak convergence to the Kesten–McKay distribution for the normalized spectral measures of random N$N$‐lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev polynomials and non‐backtracking walks.

Suggested Citation

  • Yulin Gong & Wenbo Li & Shiping Liu, 2025. "Spectral convergence of random regular graphs: Chebyshev polynomials, non‐backtracking walks, and unitary‐color extensions," Mathematische Nachrichten, Wiley Blackwell, vol. 298(10), pages 3417-3439, October.
  • Handle: RePEc:bla:mathna:v:298:y:2025:i:10:p:3417-3439
    DOI: 10.1002/mana.70046
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