26,326 research outputs found
Steinitz Theorems for Orthogonal Polyhedra
We define a simple orthogonal polyhedron to be a three-dimensional polyhedron
with the topology of a sphere in which three mutually-perpendicular edges meet
at each vertex. By analogy to Steinitz's theorem characterizing the graphs of
convex polyhedra, we find graph-theoretic characterizations of three classes of
simple orthogonal polyhedra: corner polyhedra, which can be drawn by isometric
projection in the plane with only one hidden vertex, xyz polyhedra, in which
each axis-parallel line through a vertex contains exactly one other vertex, and
arbitrary simple orthogonal polyhedra. In particular, the graphs of xyz
polyhedra are exactly the bipartite cubic polyhedral graphs, and every
bipartite cubic polyhedral graph with a 4-connected dual graph is the graph of
a corner polyhedron. Based on our characterizations we find efficient
algorithms for constructing orthogonal polyhedra from their graphs.Comment: 48 pages, 31 figure
Distance-regular graphs
This is a survey of distance-regular graphs. We present an introduction to
distance-regular graphs for the reader who is unfamiliar with the subject, and
then give an overview of some developments in the area of distance-regular
graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A.,
Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page
Contractors for flows
We answer a question raised by Lov\'asz and B. Szegedy [Contractors and
connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a
contractor for the graph parameter counting the number of B-flows of a graph,
where B is a subset of a finite Abelian group closed under inverses. We prove
our main result using the duality between flows and tensions and finite Fourier
analysis. We exhibit several examples of contractors for B-flows, which are of
interest in relation to the family of B-flow conjectures formulated by Tutte,
Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur
Magnetization in the zig-zag layered Ising model and orthogonal polynomials
We discuss the magnetization in the -th column of the zig-zag
layered 2D Ising model on a half-plane using Kadanoff-Ceva fermions and
orthogonal polynomials techniques. Our main result gives an explicit
representation of via Hankel determinants constructed from
the spectral measure of a certain Jacobi matrix which encodes the interaction
parameters between the columns. We also illustrate our approach by giving short
proofs of the classical Kaufman-Onsager-Yang and McCoy-Wu theorems in the
homogeneous setup and expressing as a Toeplitz+Hankel determinant for the
homogeneous sub-critical model in presence of a boundary magnetic field.Comment: minor updates + Section 5.3 added; 38 page
Non-coherence of arithmetic hyperbolic lattices
We prove, under the assumption of the virtual fibration conjecture for
arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n>
4, and different from 7, are non-coherent. We also establish noncoherence of
uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of
uniform lattices in SU(2,1) which have infinite abelianization.Comment: 26 pages, 3 figure
Counting cusped hyperbolic 3-manifolds that bound geometrically
We show that the number of isometry classes of cusped hyperbolic
-manifolds that bound geometrically grows at least super-exponentially with
their volume, both in the arithmetic and non-arithmetic settings.Comment: 17 pages, 7 figures; to appear in Transactions AM
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