18,397 research outputs found
Of Shadows and Gaps in Spatial Search
Spatial search occurs in a connected graph if a continuous-time quantum walk
on the adjacency matrix of the graph, suitably scaled, plus a rank-one
perturbation induced by any vertex will unitarily map the principal eigenvector
of the graph to the characteristic vector of the vertex. This phenomenon is a
natural continuous-time analogue of Grover search. The spatial search is said
to be optimal if it occurs with constant fidelity and in time inversely
proportional to the shadow of the target vertex on the principal eigenvector.
Extending a result of Chakraborty et al. (Physical Review A, 102:032214, 2020),
we prove a simpler characterization of optimal spatial search. Based on this
characterization, we observe that some families of distance-regular graphs,
such as Hamming and Grassmann graphs, have optimal spatial search. We also show
a matching lower bound on time for spatial search with constant fidelity, which
extends a bound due to Farhi and Gutmann for perfect fidelity. Our elementary
proofs employ standard tools, such as Weyl inequalities and Cauchy determinant
formula.Comment: 23 pages, 3 figure
Birthday Inequalities, Repulsion, and Hard Spheres
We study a birthday inequality in random geometric graphs: the probability of
the empty graph is upper bounded by the product of the probabilities that each
edge is absent. We show the birthday inequality holds at low densities, but
does not hold in general. We give three different applications of the birthday
inequality in statistical physics and combinatorics: we prove lower bounds on
the free energy of the hard sphere model and upper bounds on the number of
independent sets and matchings of a given size in d-regular graphs.
The birthday inequality is implied by a repulsion inequality: the expected
volume of the union of spheres of radius r around n randomly placed centers
increases if we condition on the event that the centers are at pairwise
distance greater than r. Surprisingly we show that the repulsion inequality is
not true in general, and in particular that it fails in 24-dimensional
Euclidean space: conditioning on the pairwise repulsion of centers of
24-dimensional spheres can decrease the expected volume of their union
Self-avoiding walks and connective constants
The connective constant of a quasi-transitive graph is the
asymptotic growth rate of the number of self-avoiding walks (SAWs) on from
a given starting vertex. We survey several aspects of the relationship between
the connective constant and the underlying graph .
We present upper and lower bounds for in terms of the
vertex-degree and girth of a transitive graph.
We discuss the question of whether for transitive
cubic graphs (where denotes the golden mean), and we introduce the
Fisher transformation for SAWs (that is, the replacement of vertices by
triangles).
We present strict inequalities for the connective constants
of transitive graphs , as varies.
As a consequence of the last, the connective constant of a Cayley
graph of a finitely generated group decreases strictly when a new relator is
added, and increases strictly when a non-trivial group element is declared to
be a further generator.
We describe so-called graph height functions within an account of
"bridges" for quasi-transitive graphs, and indicate that the bridge constant
equals the connective constant when the graph has a unimodular graph height
function.
A partial answer is given to the question of the locality of
connective constants, based around the existence of unimodular graph height
functions.
Examples are presented of Cayley graphs of finitely presented
groups that possess graph height functions (that are, in addition, harmonic and
unimodular), and that do not.
The review closes with a brief account of the "speed" of SAW.Comment: Accepted version. arXiv admin note: substantial text overlap with
arXiv:1304.721
The isoperimetric constant of the random graph process
The isoperimetric constant of a graph on vertices, , is the
minimum of , taken over all nonempty subsets
of size at most , where denotes the set of
edges with precisely one end in . A random graph process on vertices,
, is a sequence of graphs, where
is the edgeless graph on vertices, and
is the result of adding an edge to ,
uniformly distributed over all the missing edges. We show that in almost every
graph process equals the minimal degree of
as long as the minimal degree is . Furthermore,
we show that this result is essentially best possible, by demonstrating that
along the period in which the minimum degree is typically , the
ratio between the isoperimetric constant and the minimum degree falls from 1 to
1/2, its final value
On the extreme eigenvalues of regular graphs
In this paper, we present an elementary proof of a theorem of Serre
concerning the greatest eigenvalues of -regular graphs. We also prove an
analogue of Serre's theorem regarding the least eigenvalues of -regular
graphs: given , there exist a positive constant
and a nonnegative integer such that for any -regular graph
with no odd cycles of length less than , the number of eigenvalues
of such that is at least . This
implies a result of Winnie Li.Comment: accepted to J.Combin.Theory, Series B. added 5 new references, some
comments on the constant c in Section
Geometric inequalities in Carnot groups
Let \GG be a sub-Riemannian -step Carnot group of homogeneous dimension
. In this paper, we shall prove several geometric inequalities concerning
smooth hypersurfaces (i.e. codimension one submanifolds) immersed in \GG,
endowed with the \HH-perimeter measure.Comment: 26 page
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