20 research outputs found
The limit points of the top and bottom eigenvalues of regular graphs
We prove that for each the set of all limit points of the second
largest eigenvalue of growing sequences of -regular graphs is
. A similar argument shows that the set of all limit points of
the smallest eigenvalue of growing sequences of -regular graphs with growing
(odd) girth is . The more general question of identifying
all vectors which are limit points of the vectors of the top eigenvalues of
sequences of -regular graphs is considered as well. As a by product, in the
study of discrete counterpart of the "scarring" phenomenon observed in the
investigation of quantum ergodicity on manifolds, our technique provides a
method to construct -regular almost Ramanujan graphs with large girth and
localized eigenvectors corresponding to eigenvalues larger than ,
strengthening a result of Alon, Ganguly, and Srivastava
High-Girth Near-Ramanujan Graphs with Lossy Vertex Expansion
Kahale proved that linear sized sets in -regular Ramanujan graphs have
vertex expansion and complemented this with construction of
near-Ramanujan graphs with vertex expansion no better than .
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 for prime and infinitely many , we exhibit an
-vertex -regular graph with girth and vertex
expansion of sublinear sized sets bounded by whose nontrivial
eigenvalues are bounded in magnitude by .
2. In any Ramanujan graph with girth , all sets of size bounded by
have vertex expansion .
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
Spectrum Preserving Short Cycle Removal on Regular Graphs
We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial properties are related to the number and distance between short cycles and are known to happen with high probability in uniformly random regular graphs.
Using this method we can show two results involving high girth spectral expander graphs. First, we show that given d ? 3 and n, there exists an explicit distribution of d-regular ?(n)-vertex graphs where with high probability its samples have girth ?(log_{d-1} n) and are ?-near-Ramanujan; i.e., its eigenvalues are bounded in magnitude by 2?{d-1} + ? (excluding the single trivial eigenvalue of d). Then, for every constant d ? 3 and ? > 0, we give a deterministic poly(n)-time algorithm that outputs a d-regular graph on ?(n)-vertices that is ?-near-Ramanujan and has girth ?(?{log n}), based on the work of [Mohanty et al., 2020]
The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph
Anantharaman and Le Masson proved that any family of eigenbases of the
adjacency operators of a family of graphs is quantum ergodic (a form of
delocalization) assuming the graphs satisfy conditions of expansion and high
girth. In this paper, we show that neither of these two conditions is
sufficient by itself to necessitate quantum ergodicity. We also show that
having conditions of expansion and a specific relaxation of the high girth
constraint present in later papers on quantum ergodicity is not sufficient. We
do so by proving new properties of the Cartesian product of two graphs where
one is infinite.Comment: 12 pages, 3 figure
The necessity of conditions for graph quantum ergodicity and Cartesian products with an infinite graph
Anantharaman and Le Masson proved that any family of eigenbases of the adjacency operators of a family of graphs is quantum ergodic (a form of delocalization) assuming the graphs satisfy conditions of expansion and high girth. In this paper, we show that neither of these two conditions is sufficient by itself to necessitate quantum ergodicity. We also show that having conditions of expansion and a specific relaxation of the high girth constraint present in later papers on quantum ergodicity is not sufficient. We do so by proving new properties of the Cartesian product of two graphs where one is infinite