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

In this paper, we give a short proof of the weak convergence to the Kesten-McKay distribution for the normalized spectral measures of random $N$-lifts. This result is derived by generalizing a formula of Friedman involving Chebyshev polynomials and non-backtracking walks. We also 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 $(q_n+1)$-regular graphs $G_n$ with $n$ vertices, if $q_n=n^{o(1)}$ and $q_n$ tends to infinity, the normalized spectral measure converges almost surely in $p$-Wasserstein distance to the semicircle distribution for any $p\in [1,\mathrm{\infty })$. This strengthens a result of Dumitriu and Pal. Many of the results are extended to unitary-colored regular graphs.