Line Patterns in Free Groups

We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a line pattern in a free group acts by isometries on a related space if and only if there are no cut pairs in the decomposition space.Comment: 35 pages, 22 figures, PDFLatex; v2. finite index requires extra hypothesis; v3. 37 pages, 24 figures: updated references and add example in Section 6.3 of a rigid pattern for which the free group is not finite index in the group of pattern preserving quasi-isometries; v4. 40 pages, 26 figures: improved exposition and add example in Section 6.4 of a rigid pattern whose cube complex is not a tre

Split and join: strong partitions and Universal Steiner trees for graphs

We study the problem of constructing universal Steiner trees for undirected graphs. Given a graph G and a root node r, we seek a single spanning tree T of minimum stretch, where the stretch of T is defined to be the maximum ratio, over all subsets of terminals X, of the ratio of the cost of the sub-tree TX that connects r to X to the cost of an optimal Steiner tree connecting X to r. Universal Steiner trees (USTs) are important for data aggregation problems where computing the Steiner tree from scratch for every input instance of terminals is costly, as for example in low energy sensor network applications. We provide a polynomial time UST construction for general graphs with 2O(√log n)-stretch. We also give a polynomial time polylogarithmic-stretch construction for minor-free graphs. One basic building block in our algorithm is a hierarchy of graph partitions, each of which guarantees small strong cluster diameter and bounded local neighbourhood intersections. Our partition hierarchy for minor-free graphs is based on the solution to a cluster aggregation problem that may be of independent interest. To our knowledge, this is the first sub-linear UST result for general graphs, and the first polylogarithmic construction for minor-free graphs

Decomposition of 3-connected cubic graphs

AbstractWe solve a conjecture of Foulds and Robinson (1979) on decomposable triangulations in the plane, in the more general context of a decomposition theory of cubic 3-connected graphs. The decomposition gives us a natural way to obtain some known results about specific homeomorphic subgraphs and the extremal diameter of 3-connected cubic graphs

Diagonal Ladders: A New Class of Models for Strongly Coupled Electron Systems

We introduce a class of models defined on ladders with a diagonal structure generated by $n_p$ plaquettes. The case $n_p=1$ corresponds to the necklace ladder and has remarkable properties which are studied using DMRG and recurrent variational ansatzes. The AF Heisenberg model on this ladder is equivalent to the alternating spin-1/spin-1/2 AFH chain which is known to have a ferrimagnetic ground state (GS). For doping 1/3 the GS is a fully doped (1,1) stripe with the holes located mostly along the principal diagonal while the minor diagonals are occupied by spin singlets. This state can be seen as a Mott insulator of localized Cooper pairs on the plaquettes. A physical picture of our results is provided by a $t_p-J_p$ model of plaquettes coupled diagonally with a hopping parameter $t_d$. In the limit $t_d \to \infty$ we recover the original $t-J$ model on the necklace ladder while for weak hopping parameter the model is easily solvable. The GS in the strong hopping regime is essentially an "on link" Gutzwiller projection of the weak hopping GS. We generalize the $t_p-J_p-t_d$ model to diagonal ladders with $n_p >1$ and the 2D square lattice. We use in our construction concepts familiar in Statistical Mechanics as medial graphs and Bratelli diagrams.Comment: REVTEX file, 22 pages (twocolumn), 35 figures inserted in text. 12 Table

Graph curves

AbstractWe study a family of stable curves defined combinatorially from a trivalent graph. Most of our results are related to the conjecture of Green which relates the Clifford index of a smooth curve, an important intrinsic invariant measuring the “specialness” of the geometry of the curve, to the “resolution Clifford index,” a projective invariant defined from the canonical embedding. Thus we study the canonical linear series and its powers and identify them in terms of combinatorial data on the graph; we given combinatorial criteria for the canonical series to be base point free or very ample; we prove the analogue of Noether's theorem on the projective normality of smooth canonical curves; we define a combinatorial invariant of a graph which we conjecture to be equal to the resolution Clifford index of the associated graph curve, at least for “most” graphs; and we prove our conjecture for planar graphs and for graphs of Clifford index 0. Along the way we prove a result of some independent interest on the canonical sheaves of (not necessarily arithmetically Cohen-Macaulay) face varieties. The Appendix establishes a formula connecting the combinatorics of a trivalent graph G and the minimal degree of an admissible covering of a curve of arithmetic genus 0 by the corresponding graph curve