Brick assignments and homogeneously almost self-complementary graphs

AbstractA graph is called almost self-complementary if it is isomorphic to the graph obtained from its complement by removing a 1-factor. In this paper, we study a special class of vertex-transitive almost self-complementary graphs called homogeneously almost self-complementary. These graphs occur as factors of symmetric index-2 homogeneous factorizations of the “cocktail party graphs” K2n−nK2. We construct several infinite families of homogeneously almost self-complementary graphs, study their structure, and prove several classification results, including the characterization of all integers n of the form n=pr and n=2p with p prime for which there exists a homogeneously almost self-complementary graph on 2n vertices

Tautological relations and the r-spin Witten conjecture

In a series of two preprints, Y.-P. Lee studied relations satisfied by all formal Gromov-Witten potentials, as defined by A. Givental. He called them "universal relations" and studied their connection with tautological relations in the cohomology ring of moduli spaces of stable curves. Building on Y.-P. Lee's work, we give a simple proof of the fact that every tautological relation gives rise to a universal relation (which was also proved by Y.-P. Lee modulo certain results announced by C. Teleman). In particular, this implies that in any semi-simple Gromov-Witten theory where arbitrary correlators can be expressed in genus 0 correlators using only tautological relations, the formal and the geometric Gromov-Witten potentials coincide. As the most important application, we show that our results suffice to deduce the statement of a 1991 Witten conjecture on r-spin structures from the results obtained by Givental for the corresponding formal Gromov-Witten potential. The conjecture in question states that certain intersection numbers on the moduli space of r-spin structures can be arranged into a power series that satisfies the r-KdV (or r-th higher Gelfand-Dikii) hierarchy of partial differential equations.Comment: 46 pages, 7 figures, A discussion of the analyticity of Gromov-Witten potentials and a more careful description of Givental's group action added in Section 5; minor changes elsewher

From rubber bands to rational maps: A research report

This research report outlines work, partially joint with Jeremy Kahn and Kevin Pilgrim, which gives parallel theories of elastic graphs and conformal surfaces with boundary. One one hand, this lets us tell when one rubber band network is looser than another, and on the other hand tell when one conformal surface embeds in another. We apply this to give a new characterization of hyperbolic critically finite rational maps among branched self-coverings of the sphere, by a positive criterion: a branched covering is equivalent to a hyperbolic rational map if and only if there is an elastic graph with a particular "self-embedding" property. This complements the earlier negative criterion of W. Thurston.Comment: 52 pages, numerous figures. v2: New example

Analysis of existing mathematics textbooks for use in secondary schools.

Thesis (Ed.M.)--Boston University Thesis (M.A.)--Boston Universit

Some colouring problems for Paley graphs

The Paley graph Pq, where q≡1(mod4) is a prime power, is the graph with vertices the elements of the finite field Fq and an edge between x and y if and only if x-y is a non-zero square in Fq. This paper gives new results on some colouring problems for Paley graphs and related discussion. © 2005 Elsevier B.V. All rights reserved

Minimal vertex covers on finite-connectivity random graphs - a hard-sphere lattice-gas picture

The minimal vertex-cover (or maximal independent-set) problem is studied on random graphs of finite connectivity. Analytical results are obtained by a mapping to a lattice gas of hard spheres of (chemical) radius one, and they are found to be in excellent agreement with numerical simulations. We give a detailed description of the replica-symmetric phase, including the size and the entropy of the minimal vertex covers, and the structure of the unfrozen component which is found to percolate at connectivity $c\simeq 1.43$. The replica-symmetric solution breaks down at $c=e\simeq 2.72$. We give a simple one-step replica symmetry broken solution, and discuss the problems in interpretation and generalization of this solution.Comment: 32 pages, 9 eps figures, to app. in PRE (01 May 2001

Flipping Cubical Meshes

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th International Meshing Roundtable. This version removes some unwanted paragraph breaks from the previous version; the text is unchange