Representations of Circular Words

In this article we give two different ways of representations of circular words. Representations with tuples are intended as a compact notation, while representations with trees give a way to easily process all conjugates of a word. The latter form can also be used as a graphical representation of periodic properties of finite (in some cases, infinite) words. We also define iterative representations which can be seen as an encoding utilizing the flexible properties of circular words. Every word over the two letter alphabet can be constructed starting from ab by applying the fractional power and the cyclic shift operators one after the other, iteratively.Comment: In Proceedings AFL 2014, arXiv:1405.527

Tur\'an Graphs, Stability Number, and Fibonacci Index

The Fibonacci index of a graph is the number of its stable sets. This parameter is widely studied and has applications in chemical graph theory. In this paper, we establish tight upper bounds for the Fibonacci index in terms of the stability number and the order of general graphs and connected graphs. Tur\'an graphs frequently appear in extremal graph theory. We show that Tur\'an graphs and a connected variant of them are also extremal for these particular problems.Comment: 11 pages, 3 figure

Heisenberg antiferromagnet on Cayley trees: low-energy spectrum and even/odd site imbalance

To understand the role of local sublattice imbalance in low-energy spectra of s=1/2 quantum antiferromagnets, we study the s=1/2 quantum nearest neighbor Heisenberg antiferromagnet on the coordination 3 Cayley tree. We perform many-body calculations using an implementation of the density matrix renormalization group (DMRG) technique for generic tree graphs. We discover that the bond-centered Cayley tree has a quasidegenerate set of a low-lying tower of states and an "anomalous" singlet-triplet finite-size gap scaling. For understanding the construction of the first excited state from the many-body ground state, we consider a wave function ansatz given by the single-mode approximation, which yields a high overlap with the DMRG wave function. Observing the ground-state entanglement spectrum leads us to a picture of the low-energy degrees of freedom being "giant spins" arising out of sublattice imbalance, which helps us analytically understand the scaling of the finite-size spin gap. The Schwinger-boson mean-field theory has been generalized to nonuniform lattices, and ground states have been found which are spatially inhomogeneous in the mean-field parameters.Comment: 19 pages, 12 figures, 6 tables. Changes made to manuscript after referee suggestions: parts reorganized, clarified discussion on Fibonacci tree, typos correcte

Growth rate for the expected value of a generalized random Fibonacci sequence

A random Fibonacci sequence is defined by the relation g_n = | g_{n-1} +/- g_{n-2} |, where the +/- sign is chosen by tossing a balanced coin for each n. We generalize these sequences to the case when the coin is unbalanced (denoting by p the probability of a +), and the recurrence relation is of the form g_n = |\lambda g_{n-1} +/- g_{n-2} |. When \lambda >=2 and 0 < p <= 1, we prove that the expected value of g_n grows exponentially fast. When \lambda = \lambda_k = 2 cos(\pi/k) for some fixed integer k>2, we show that the expected value of g_n grows exponentially fast for p>(2-\lambda_k)/4 and give an algebraic expression for the growth rate. The involved methods extend (and correct) those introduced in a previous paper by the second author