Non-backtracking eigenvector delocalization for random regular graphs

Xiangyi Zhu, Yizhe Zhu

Published 2023-12-06Version 1

The non-backtracking operator of a graph is a powerful tool in spectral graph theory and random matrix theory. Most existing results for the non-backtracking operator of a random graph concern only eigenvalues or top eigenvectors. In this paper, we take the first step in analyzing its bulk eigenvector behaviors. We demonstrate that for the non-backtracking operator $B$ of a random $d$-regular graph, its eigenvectors corresponding to nontrivial eigenvalues are completely delocalized with high probability. Additionally, we show complete delocalization for a reduced $2n \times 2n$ non-backtracking matrix $\widetilde{B}$. By projecting all eigenvalues of $\widetilde{B}$ onto the real line, we obtain an empirical measure that converges weakly in probability to the Kesten-McKay law for fixed $d\geq 3$ and to a semicircle law as $d \to\infty$ with $n \to\infty$.