18,800 research outputs found
A transfer principle and applications to eigenvalue estimates for graphs
In this paper, we prove a variant of the Burger-Brooks transfer principle
which, combined with recent eigenvalue bounds for surfaces, allows to obtain
upper bounds on the eigenvalues of graphs as a function of their genus. More
precisely, we show the existence of a universal constants such that the
-th eigenvalue of the normalized Laplacian of a graph
of (geometric) genus on vertices satisfies where denotes the maximum valence of
vertices of the graph. This result is tight up to a change in the value of the
constant , and improves recent results of Kelner, Lee, Price and Teng on
bounded genus graphs.
To show that the transfer theorem might be of independent interest, we relate
eigenvalues of the Laplacian on a metric graph to the eigenvalues of its simple
graph models, and discuss an application to the mesh partitioning problem,
extending pioneering results of Miller-Teng-Thurston-Vavasis and Spielman-Tang
to arbitrary meshes.Comment: Major revision, 16 page
Shortest path embeddings of graphs on surfaces
The classical theorem of F\'{a}ry states that every planar graph can be
represented by an embedding in which every edge is represented by a straight
line segment. We consider generalizations of F\'{a}ry's theorem to surfaces
equipped with Riemannian metrics. In this setting, we require that every edge
is drawn as a shortest path between its two endpoints and we call an embedding
with this property a shortest path embedding. The main question addressed in
this paper is whether given a closed surface S, there exists a Riemannian
metric for which every topologically embeddable graph admits a shortest path
embedding. This question is also motivated by various problems regarding
crossing numbers on surfaces.
We observe that the round metrics on the sphere and the projective plane have
this property. We provide flat metrics on the torus and the Klein bottle which
also have this property.
Then we show that for the unit square flat metric on the Klein bottle there
exists a graph without shortest path embeddings. We show, moreover, that for
large g, there exist graphs G embeddable into the orientable surface of genus
g, such that with large probability a random hyperbolic metric does not admit a
shortest path embedding of G, where the probability measure is proportional to
the Weil-Petersson volume on moduli space.
Finally, we construct a hyperbolic metric on every orientable surface S of
genus g, such that every graph embeddable into S can be embedded so that every
edge is a concatenation of at most O(g) shortest paths.Comment: 22 pages, 11 figures: Version 3 is updated after comments of
reviewer
Topological recursion relations from Pixton's formula
For any genus g \leq 26, and for n \leq 3 in all genus, we prove that every
degree-g polynomial in the psi-classes on Mbar_{g,n} can be expressed as a sum
of tautological classes supported on the boundary with no kappa-classes. Such
equations, which we refer to as topological recursion relations, can be used to
deduce universal equations for the Gromov-Witten invariants of any target.Comment: 17 page
Skeletons of stable maps II: Superabundant geometries
We implement new techniques involving Artin fans to study the realizability
of tropical stable maps in superabundant combinatorial types. Our approach is
to understand the skeleton of a fundamental object in logarithmic
Gromov--Witten theory -- the stack of prestable maps to the Artin fan. This is
used to examine the structure of the locus of realizable tropical curves and
derive 3 principal consequences. First, we prove a realizability theorem for
limits of families of tropical stable maps. Second, we extend the sufficiency
of Speyer's well-spacedness condition to the case of curves with good
reduction. Finally, we demonstrate the existence of liftable genus 1
superabundant tropical curves that violate the well-spacedness condition.Comment: 17 pages, 1 figure. v2 fixes a minor gap in the proof of Theorem D
and adds details to the construction of the skeleton of a toroidal Artin
stack. Minor clarifications throughout. To appear in Research in the
Mathematical Science
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