Generalized Fibonacci cubes

AbstractGeneralized Fibonacci cube Qd(f) is introduced as the graph obtained from the d-cube Qd by removing all vertices that contain a given binary string f as a substring. In this notation, the Fibonacci cube Γd is Qd(11). The question whether Qd(f) is an isometric subgraph of Qd is studied. Embeddable and non-embeddable infinite series are given. The question is completely solved for strings f of length at most five and for strings consisting of at most three blocks. Several properties of the generalized Fibonacci cubes are deduced. Fibonacci cubes are, besides the trivial cases Qd(10) and Qd(01), the only generalized Fibonacci cubes that are median closed subgraphs of the corresponding hypercubes. For admissible strings f, the f-dimension of a graph is introduced. Several problems and conjectures are also listed

Unconventional phase transitions in random systems

In this thesis we study the effects of different types of disorder and quasiperiodic modulations on quantum, classical and nonequilibrium phase transitions. After a brief introduction, we examine the effect of topological disorder on phase transitions and explain a host of violations of the Harris and Imry-Ma criteria that predict the fate of disordered phase transitions. We identify a class of random and quasiperiodic lattices in which a topological constraint introduces strong anticorrelations leading to modifications of the Harris and Imry-Ma criteria for such lattices. We investigate whether or not the Imry-Ma criterion, that predicts that random field disorder destroys phase transitions in equilibrium systems in sufficiently low dimensions, also holds for nonequilibrium phase transitions. We find that the Imry-Ma criterion does not apply to a prototypical absorbing state nonequilibrium transition. In addition, we study the effect of disorder with long-range spatial correlations on the absorbing state phase transition in the contact process. Most importantly, we find that long-range correlations enhance the Griffiths singularities and change the universality class of the transition. We also investigate the absorbing state phase transition of the contact process with quasiperiodic transition rates using a real-space renormalization group which yields a complete theory of the resulting exotic infinite-modulation critical point. Moreover, we study the effect of quenched disorder on a randomly layered Heisenberg magnet by means of a large-scale Monte-Carlo simulations. We find that the transition follows the infinite-randomness critical point scenario. Finally, we investigate the effect of quenched disorder on the first-order phase transition in the N-color quantum Ashkin-Teller model by means of strong-disorder renormalization group theory. We find that disorder rounds the first-order quantum phase transition in agreement with quantum version of the Imry-Ma criterion --Abstract, page v

Phonons in disordered harmonic lattices

This work explores the nature of the normal modes of vibration for harmonic lattices with the inclusion of disorder in one-dimension (1D) and three-dimensions (3D). The model systems can be visualised as a `ball' and `spring' model in simple cubic configuration, and the disorder is applied to the magnitudes of the masses, or the force constants of the interatomic `springs' in the system. With the analogous nature between the electronic tight binding Hamiltonian for potential disordered electronic systems and the isotropic Born model for phonons in mass disordered lattices we analyse in detail a transformation between the normal modes of vibration throughout a mass disordered harmonic lattice and the electron wave function of the tight-binding Hamiltonian. The transformation is applied to density of states (DOS) calculations and is also particularly useful for determining the phase diagrams for the phonon localisation-delocalisation transition (LDT). The LDT phase boundary for the spring constant disordered system is obtained with good resolution and the mass disordered phase boundary is verified with high precision transfer-matrix method (TMM) results. High accuracy critical parameters are obtained for three transitions for each type of disorder by finite size scaling (FSS), and consequently the critical exponent that characterises the transition is found as = 1:550+0:020 -0:017 which indicates that the transition is of the same orthogonal universality class as the electronic Anderson transition. With multifractal analysis of the generalised inverse participation ratio (gIPR) for the critical transition frequency states at spring constant disorder width k = 10 and mass disorder width m = 1:2 we confirm that the singularity spectrum is the same within error as the electronic singularity spectrum at criticality and can be considered to be universal. We further investigate the nature of the modes throughout the spectrum of the disordered systems with vibrational eigenstate statistics. We find deviations of the vibrational displacement uctuations away from the Porter-Thomas distribution (PTD) and show that the deviations are within the vicinity of the so called `bosonpeak' (BP) indicating the possible significance of the BP

Shortest Paths in Graphs of Convex Sets

Given a graph, the shortest-path problem requires finding a sequence of edges with minimum cumulative length that connects a source to a target vertex. We consider a generalization of this classical problem in which the position of each vertex in the graph is a continuous decision variable, constrained to lie in a corresponding convex set. The length of an edge is then defined as a convex function of the positions of the vertices it connects. Problems of this form arise naturally in road networks, robot navigation, and even optimal control of hybrid dynamical systems. The price for such a wide applicability is the complexity of this problem, which is easily seen to be NP-hard. Our main contribution is a strong mixed-integer convex formulation based on perspective functions. This formulation has a very tight convex relaxation and allows to efficiently find globally-optimal paths in large graphs and in high-dimensional spaces

An extensive English language bibliography on graph theory and its applications

Bibliography on graph theory and its application