A New Quartet Tree Heuristic for Hierarchical Clustering

We consider the problem of constructing an an optimal-weight tree from the 3*(n choose 4) weighted quartet topologies on n objects, where optimality means that the summed weight of the embedded quartet topologiesis optimal (so it can be the case that the optimal tree embeds all quartets as non-optimal topologies). We present a heuristic for reconstructing the optimal-weight tree, and a canonical manner to derive the quartet-topology weights from a given distance matrix. The method repeatedly transforms a bifurcating tree, with all objects involved as leaves, achieving a monotonic approximation to the exact single globally optimal tree. This contrasts to other heuristic search methods from biological phylogeny, like DNAML or quartet puzzling, which, repeatedly, incrementally construct a solution from a random order of objects, and subsequently add agreement values.Comment: 22 pages, 14 figure

Simulating Strongly Correlated Quantum Systems with Tree Tensor Networks

We present a tree-tensor-network-based method to study strongly correlated systems with nonlocal interactions in higher dimensions. Although the momentum-space and quantum-chemistry versions of the density matrix renormalization group (DMRG) method have long been applied to such systems, the spatial topology of DMRG-based methods allows efficient optimizations to be carried out with respect to one spatial dimension only. Extending the matrix-product-state picture, we formulate a more general approach by allowing the local sites to be coupled to more than two neighboring auxiliary subspaces. Following Shi. et. al. [Phys. Rev. A, 74, 022320 (2006)], we treat a tree-like network ansatz with arbitrary coordination number z, where the z=2 case corresponds to the one-dimensional scheme. For this ansatz, the long-range correlation deviates from the mean-field value polynomially with distance, in contrast to the matrix-product ansatz, which deviates exponentially. The computational cost of the tree-tensor-network method is significantly smaller than that of previous DMRG-based attempts, which renormalize several blocks into a single block. In addition, we investigate the effect of unitary transformations on the local basis states and present a method for optimizing such transformations. For the 1-d interacting spinless fermion model, the optimized transformation interpolates smoothly between real space and momentum space. Calculations carried out on small quantum chemical systems support our approach

Distance matrices of a tree: two more invariants, and in a unified framework

Graham-Pollak showed that for $D = D_T$ the distance matrix of a tree $T$, det$(D)$ depends only on its number of edges. Several other variants of $D$, including directed/multiplicative/$q$- versions were studied, and always, det$(D)$ depends only on the edge-data. We introduce a general framework for bi-directed weighted trees, with threefold significance. First, we improve on state-of-the-art for all known variants, even in the classical Graham-Pollak case: we delete arbitrary pendant nodes (and more general subsets) from the rows/columns of $D$, and show these minors do not depend on the tree-structure. Second, our setting unifies all known variants (with entries in a commutative ring). We further compute in closed form the inverse of $D$, extending a result of Graham-Lovasz [Adv. Math. 1978] and answering a question of Bapat-Lal-Pati [Lin. Alg. Appl. 2006]. Third, we compute a second function of the matrix $D$: the sum of all its cofactors, cof$(D)$. This was worked out in the simplest setting by Graham-Hoffman-Hosoya (1978), but is relatively unexplored for other variants. We prove a stronger result, in our general setting, by computing cof$(.)$ for minors as above, and showing these too depend only on the edge-data. Finally, we show our setting is the 'most general possible', in that with more freedom in the edgeweights, det$(D)$ and cof$(D)$ depend on the tree structure. In a sense, this completes the study of the invariant det$(D_T)$ (and cof$(D_T)$) for trees $T$ with edge-data in a commutative ring. Moreover: for a bi-directed graph $G$ we prove multiplicative Graham-Hoffman-Hosoya type formulas for det$(D_G)$, cof$(D_G)$, $D_G^{-1}$. We then show how this subsumes their 1978 result. The final section introduces and computes a third, novel invariant for trees and a Graham-Hoffman-Hosoya type result for our "most general" distance matrix $D_T$.Comment: 42 pages, 2 figures; minor edits in the proof of Theorems A and 1.1

Quantum field theory on quantum graphs and application to their conductance

We construct a bosonic quantum field on a general quantum graph. Consistency of the construction leads to the calculation of the total scattering matrix of the graph. This matrix is equivalent to the one already proposed using generalized star product approach. We give several examples and show how they generalize some of the scattering matrices computed in the mathematical or condensed matter physics litterature. Then, we apply the construction for the calculation of the conductance of graphs, within a small distance approximation. The consistency of the approximation is proved by direct comparison with the exact calculation for the `tadpole' graph.Comment: 32 pages; misprints in tree graph corrected; proofs of consistency and unitarity adde