1,444 research outputs found
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 class of graphs with strong mixing properties
We study three mixing properties of a graph: large algebraic connectivity,
large Cheeger constant (isoperimetric number) and large spectral gap from 1 for
the second largest eigenvalue of the transition probability matrix of the
random walk on the graph. We prove equivalence of this properties (in some
sense). We give estimates for the probability for a random graph to satisfy
these properties. In addition, we present asymptotic formulas for the numbers
of Eulerian orientations and Eulerian circuits in an undirected simple graph
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Isoperimetric Inequalities on Hexagonal Grids
We consider the edge- and vertex-isoperimetric probem on finite and infinite
hexagonal grids: For a subset W of the hexagonal grid of given cardinality, we
give a lower bound for the number of edges between W and its complement, and
lower bounds for the number of vertices in the neighborhood of W and for the
number of vertices in the boundary of W. For the infinite hexagonal grid the
given bounds are tight
Unimodular graphs and Eisenstein sums
Motivated in part by combinatorial applications to certain sum-product
phenomena, we introduce unimodular graphs over finite fields and, more
generally, over finite valuation rings. We compute the spectrum of the
unimodular graphs, by using Eisenstein sums associated to unramified extensions
of such rings. We derive an estimate for the number of solutions to the
restricted dot product equation over a finite valuation ring.
Furthermore, our spectral analysis leads to the exact value of the
isoperimetric constant for half of the unimodular graphs. We also compute the
spectrum of Platonic graphs over finite valuation rings, and products of such
rings - e.g., . In particular, we deduce an improved lower
bound for the isoperimetric constant of the Platonic graph over
.Comment: V2: minor revisions. To appear in the Journal of Algebraic
Combinatoric
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