325 research outputs found
Ramanujan Coverings of Graphs
Let be a finite connected graph, and let be the spectral radius of
its universal cover. For example, if is -regular then
. We show that for every , there is an -covering
(a.k.a. an -lift) of where all the new eigenvalues are bounded from
above by . It follows that a bipartite Ramanujan graph has a Ramanujan
-covering for every . This generalizes the case due to Marcus,
Spielman and Srivastava (2013).
Every -covering of corresponds to a labeling of the edges of by
elements of the symmetric group . We generalize this notion to labeling
the edges by elements of various groups and present a broader scenario where
Ramanujan coverings are guaranteed to exist.
In particular, this shows the existence of richer families of bipartite
Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava,
a crucial component of our proof is the existence of interlacing families of
polynomials for complex reflection groups. The core argument of this component
is taken from a recent paper of them (2015).
Another important ingredient of our proof is a new generalization of the
matching polynomial of a graph. We define the -th matching polynomial of
to be the average matching polynomial of all -coverings of . We show this
polynomial shares many properties with the original matching polynomial. For
example, it is real rooted with all its roots inside .Comment: 38 pages, 4 figures, journal version (minor changes from previous
arXiv version). Shortened version appeared in STOC 201
Dimers, Tilings and Trees
Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others
we describe a natural equivalence between three planar objects: weighted
bipartite planar graphs; planar Markov chains; and tilings with convex
polygons. This equivalence provides a measure-preserving bijection between
dimer coverings of a weighted bipartite planar graph and spanning trees on the
corresponding Markov chain. The tilings correspond to harmonic functions on the
Markov chain and to ``discrete analytic functions'' on the bipartite graph.
The equivalence is extended to infinite periodic graphs, and we classify the
resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure
On the Expansion of Group-Based Lifts
A -lift of an -vertex base graph is a graph on
vertices, where each vertex of is replaced by vertices
and each edge in is replaced by a matching
representing a bijection so that the edges of are of the form
. Lifts have been studied as a means to efficiently
construct expanders. In this work, we study lifts obtained from groups and
group actions. We derive the spectrum of such lifts via the representation
theory principles of the underlying group. Our main results are:
(1) There is a constant such that for every , there
does not exist an abelian -lift of any -vertex -regular base graph
with being almost Ramanujan (nontrivial eigenvalues of the adjacency matrix
at most in magnitude). This can be viewed as an analogue of the
well-known no-expansion result for abelian Cayley graphs.
(2) A uniform random lift in a cyclic group of order of any -vertex
-regular base graph , with the nontrivial eigenvalues of the adjacency
matrix of bounded by in magnitude, has the new nontrivial
eigenvalues also bounded by in magnitude with probability
. In particular, there is a constant such that for
every , there exists a lift of every Ramanujan graph in
a cyclic group of order with being almost Ramanujan. We use this to
design a quasi-polynomial time algorithm to construct almost Ramanujan
expanders deterministically.
The existence of expanding lifts in cyclic groups of order
can be viewed as a lower bound on the order of the largest abelian group
that produces expanding lifts. Our results show that the lower bound matches
the upper bound for (upto in the exponent)
- β¦