Kahale proved that linear sized sets in d-regular Ramanujan graphs have
vertex expansion ∼2d and complemented this with construction of
near-Ramanujan graphs with vertex expansion no better than 2d.
However, the construction of Kahale encounters highly local obstructions to
better vertex expansion. In particular, the poorly expanding sets are
associated with short cycles in the graph. Thus, it is natural to ask whether
high-girth Ramanujan graphs have improved vertex expansion. Our results are
two-fold:
1. For every d=p+1 for prime p and infinitely many n, we exhibit an
n-vertex d-regular graph with girth Ω(logd−1n) and vertex
expansion of sublinear sized sets bounded by 2d+1 whose nontrivial
eigenvalues are bounded in magnitude by 2d−1+O(logn1).
2. In any Ramanujan graph with girth Clogn, all sets of size bounded by
n0.99C/4 have vertex expansion (1−od(1))d.
The tools in analyzing our construction include the nonbacktracking operator
of an infinite graph, the Ihara--Bass formula, a trace moment method inspired
by Bordenave's proof of Friedman's theorem, and a method of Kahale to study
dispersion of eigenvalues of perturbed graphs.Comment: 15 pages, 1 figur