2,664 research outputs found
An entropy based proof of the Moore bound for irregular graphs
We provide proofs of the following theorems by considering the entropy of
random walks: Theorem 1.(Alon, Hoory and Linial) Let G be an undirected simple
graph with n vertices, girth g, minimum degree at least 2 and average degree d:
Odd girth: If g=2r+1,then n \geq 1 + d*(\Sum_{i=0}^{r-1}(d-1)^i) Even girth: If
g=2r,then n \geq 2*(\Sum_{i=0}^{r-1} (d-1)^i) Theorem 2.(Hoory) Let G =
(V_L,V_R,E) be a bipartite graph of girth g = 2r, with n_L = |V_L| and n_R =
|V_R|, minimum degree at least 2 and the left and right average degrees d_L and
d_R. Then, n_L \geq \Sum_{i=0}^{r-1}(d_R-1)^{i/2}(d_L-1)^{i/2} n_R \geq
\Sum_{i=0}^{r-1}(d_L-1)^{i/2}(d_R-1)^{i/2}Comment: 6 page
A proof of the Upper Matching Conjecture for large graphs
We prove that the `Upper Matching Conjecture' of Friedland, Krop, and
Markstr\"om and the analogous conjecture of Kahn for independent sets in
regular graphs hold for all large enough graphs as a function of the degree.
That is, for every and every large enough divisible by , a union of
copies of the complete -regular bipartite graph maximizes the
number of independent sets and matchings of size for each over all
-regular graphs on vertices. To prove this we utilize the cluster
expansion for the canonical ensemble of a statistical physics spin model, and
we give some further applications of this method to maximizing and minimizing
the number of independent sets and matchings of a given size in regular graphs
of a given minimum girth
A simple and sharper proof of the hypergraph Moore bound
The hypergraph Moore bound is an elegant statement that characterizes the
extremal trade-off between the girth - the number of hyperedges in the smallest
cycle or even cover (a subhypergraph with all degrees even) and size - the
number of hyperedges in a hypergraph. For graphs (i.e., -uniform
hypergraphs), a bound tight up to the leading constant was proven in a
classical work of Alon, Hoory and Linial [AHL02]. For hypergraphs of uniformity
, an appropriate generalization was conjectured by Feige [Fei08]. The
conjecture was settled up to an additional factor in the size
in a recent work of Guruswami, Kothari and Manohar [GKM21]. Their argument
relies on a connection between the existence of short even covers and the
spectrum of a certain randomly signed Kikuchi matrix. Their analysis,
especially for the case of odd , is significantly complicated.
In this work, we present a substantially simpler and shorter proof of the
hypergraph Moore bound. Our key idea is the use of a new reweighted Kikuchi
matrix and an edge deletion step that allows us to drop several involved steps
in [GKM21]'s analysis such as combinatorial bucketing of rows of the Kikuchi
matrix and the use of the Schudy-Sviridenko polynomial concentration. Our
simpler proof also obtains tighter parameters: in particular, the argument
gives a new proof of the classical Moore bound of [AHL02] with no loss (the
proof in [GKM21] loses a factor), and loses only a single
logarithmic factor for all -uniform hypergraphs.
As in [GKM21], our ideas naturally extend to yield a simpler proof of the
full trade-off for strongly refuting smoothed instances of constraint
satisfaction problems with similarly improved parameters
Quantum ergodicity for quantum graphs without back-scattering
We give an estimate of the quantum variance for -regular graphs quantised
with boundary scattering matrices that prohibit back-scattering. For families
of graphs that are expanders, with few short cycles, our estimate leads to
quantum ergodicity for these families of graphs. Our proof is based on a
uniform control of an associated random walk on the bonds of the graph. We show
that recent constructions of Ramanujan graphs, and asymptotically almost
surely, random -regular graphs, satisfy the necessary conditions to conclude
that quantum ergodicity holds.Comment: 28 pages, 5 figure
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