497 research outputs found
Distance-regular Cayley graphs with small valency
We consider the problem of which distance-regular graphs with small valency
are Cayley graphs. We determine the distance-regular Cayley graphs with valency
at most , the Cayley graphs among the distance-regular graphs with known
putative intersection arrays for valency , and the Cayley graphs among all
distance-regular graphs with girth and valency or . We obtain that
the incidence graphs of Desarguesian affine planes minus a parallel class of
lines are Cayley graphs. We show that the incidence graphs of the known
generalized hexagons are not Cayley graphs, and neither are some other
distance-regular graphs that come from small generalized quadrangles or
hexagons. Among some ``exceptional'' distance-regular graphs with small
valency, we find that the Armanios-Wells graph and the Klein graph are Cayley
graphs.Comment: 19 pages, 4 table
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¦