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We show that for every prime $d$ and $alphain (0,1/6)$, there is an infinite sequence of $(d+1)$-regular graphs $G=(V,E)$ with girth at least $2alpha log_{d}(|V|)(1-o_d(1))$, second adjacency matrix eigenvalue bounded by $(3/sqrt{2})sqrt{d}$, and many eigenvectors fully localized on small sets of size $O(|V|^alpha)$. This strengthens the results of Ganguly-Srivastava, who constructed high girth (but not expanding) graphs with similar properties, and may be viewed as a discrete analogue of the scarring phenomenon observed in the study of quantum ergodicity on manifolds. Key ingredients in the proof are a technique of Kahale for bounding the growth rate of eigenfunctions of graphs, discovered in the context of vertex expansion and a method of ErdH{o}s and Sachs for constructing high girth regular graphs.
Let $X$ be an infinite graph of bounded degree; e.g., the Cayley graph of a free product of finite groups. If $G$ is a finite graph covered by $X$, it is said to be $X$-Ramanujan if its second-largest eigenvalue $lambda_2(G)$ is at most the spectral
Let $p(Y_1, dots, Y_d, Z_1, dots, Z_e)$ be a self-adjoint noncommutative polynomial, with coefficients from $mathbb{C}^{r times r}$, in the indeterminates $Y_1, dots, Y_d$ (considered to be self-adjoint), the indeterminates $Z_1, dots, Z_e$, and thei
For every constant $d geq 3$ and $epsilon > 0$, we give a deterministic $mathrm{poly}(n)$-time algorithm that outputs a $d$-regular graph on $Theta(n)$ vertices that is $epsilon$-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by $2sqr
The r-th power of a graph modifies a graph by connecting every vertex pair within distance r. This paper gives a generalization of the Alon-Boppana Theorem for the r-th power of graphs, including irregular graphs. This leads to a generalized notion o
Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown