Direct evaluation of pure graph state entanglement

We address the question of quantifying entanglement in pure graph states. Evaluation of multipartite entanglement measures is extremely hard for most pure quantum states. In this paper we demonstrate how solving one problem in graph theory, namely the identification of maximum independent set, allows us to evaluate three multipartite entanglement measures for pure graph states. We construct the minimal linear decomposition into product states for a large group of pure graph states, allowing us to evaluate the Schmidt measure. Furthermore we show that computation of distance-like measures such as relative entropy of entanglement and geometric measure becomes tractable for these states by explicit construction of closest separable and closest product states respectively. We show how these separable states can be described using stabiliser formalism as well as PEPs-like construction. Finally we discuss the way in which introducing noise to the system can optimally destroy entanglement.Comment: 23 pages, 9 figure

Finding combinatorial structures

In this thesis we answer questions in two related areas of combinatorics: Ramsey theory and asymptotic enumeration. In Ramsey theory we introduce a new method for finding desired structures. We find a new upper bound on the Ramsey number of a path against a kth power of a path. Using our new method and this result we obtain a new upper bound on the Ramsey number of the kth power of a long cycle. As a corollary we show that, while graphs on n vertices with maximum degree k may in general have Ramsey numbers as large as ckn, if the stronger restriction that the bandwidth should be at most k is given, then the Ramsey numbers are bounded by the much smaller value. We go on to attack an old conjecture of Lehel: by using our new method we can improve on a result of Luczak, Rodl and Szemeredi [60]. Our new method replaces their use of the Regularity Lemma, and allows us to prove that for any n > 218000, whenever the edges of the complete graph on n vertices are two-coloured there exist disjoint monochromatic cycles covering all n vertices. In asymptotic enumeration we examine first the class of bipartite graphs with some forbidden induced subgraph H. We obtain some results for every H, with special focus on the cases where the growth speed of the class is factorial, and make some comments on a connection to clique-width. We then move on to a detailed discussion of 2-SAT functions. We find the correct asymptotic formula for the number of 2-SAT functions on n variables (an improvement on a result of Bollob´as, Brightwell and Leader [13], who found the dominant term in the exponent), the first error term for this formula, and some bounds on smaller error terms. Finally we obtain various expected values in the uniform model of random 2-SAT functions

Partitioning a graph into disjoint cliques and a triangle-free graph

A graph G=(V,E) is partitionable if there exists a partition {A,B} of V such that A induces a disjoint union of cliques (i.e., G[A] is P_3-free) and B induces a triangle-free graph (i.e., G[B] is K_3-free). In this paper we investigate the computational complexity of deciding whether a graph is partitionable. The problem is known to be NP-complete on arbitrary graphs. Here it is proved that if a graph G is bull-free, planar, perfect, K_4-free or does not contain certain holes then deciding whether G is partitionable is NP-complete. This answers an open question posed by Thomassé, Trotignon and Vušković. In contrast a finite list of forbidden induced subgraphs is given for partitionable cographs

Data Mining Using the Crossing Minimization Paradigm

Our ability and capacity to generate, record and store multi-dimensional, apparently unstructured data is increasing rapidly, while the cost of data storage is going down. The data recorded is not perfect, as noise gets introduced in it from different sources. Some of the basic forms of noise are incorrect recording of values and missing values. The formal study of discovering useful hidden information in the data is called Data Mining. Because of the size, and complexity of the problem, practical data mining problems are best attempted using automatic means. Data Mining can be categorized into two types i.e. supervised learning or classification and unsupervised learning or clustering. Clustering only the records in a database (or data matrix) gives a global view of the data and is called one-way clustering. For a detailed analysis or a local view, biclustering or co-clustering or two-way clustering is required involving the simultaneous clustering of the records and the attributes. In this dissertation, a novel fast and white noise tolerant data mining solution is proposed based on the Crossing Minimization (CM) paradigm; the solution works for one-way as well as two-way clustering for discovering overlapping biclusters. For decades the CM paradigm has traditionally been used for graph drawing and VLSI (Very Large Scale Integration) circuit design for reducing wire length and congestion. The utility of the proposed technique is demonstrated by comparing it with other biclustering techniques using simulated noisy, as well as real data from Agriculture, Biology and other domains. Two other interesting and hard problems also addressed in this dissertation are (i) the Minimum Attribute Subset Selection (MASS) problem and (ii) Bandwidth Minimization (BWM) problem of sparse matrices. The proposed CM technique is demonstrated to provide very convincing results while attempting to solve the said problems using real public domain data. Pakistan is the fourth largest supplier of cotton in the world. An apparent anomaly has been observed during 1989-97 between cotton yield and pesticide consumption in Pakistan showing unexpected periods of negative correlation. By applying the indigenous CM technique for one-way clustering to real Agro-Met data (2001-2002), a possible explanation of the anomaly has been presented in this thesis

Ramsey expansions of metrically homogeneous graphs

We discuss the Ramsey property, the existence of a stationary independence relation and the coherent extension property for partial isometries (coherent EPPA) for all classes of metrically homogeneous graphs from Cherlin's catalogue, which is conjectured to include all such structures. We show that, with the exception of tree-like graphs, all metric spaces in the catalogue have precompact Ramsey expansions (or lifts) with the expansion property. With two exceptions we can also characterise the existence of a stationary independence relation and the coherent EPPA. Our results can be seen as a new contribution to Ne\v{s}et\v{r}il's classification programme of Ramsey classes and as empirical evidence of the recent convergence in techniques employed to establish the Ramsey property, the expansion (or lift or ordering) property, EPPA and the existence of a stationary independence relation. At the heart of our proof is a canonical way of completing edge-labelled graphs to metric spaces in Cherlin's classes. The existence of such a "completion algorithm" then allows us to apply several strong results in the areas that imply EPPA and respectively the Ramsey property. The main results have numerous corollaries on the automorphism groups of the Fra\"iss\'e limits of the classes, such as amenability, unique ergodicity, existence of universal minimal flows, ample generics, small index property, 21-Bergman property and Serre's property (FA).Comment: 57 pages, 14 figures. Extends results of arXiv:1706.00295. Minor revisio

Algorithms for exploring structure in complex networks

As real-world networks grow increasingly complex, new and adaptable methods are required to analyse and understand the defining features of these networks. To find and exploit these unique network features, current methods, primarily for random model generation and detection of anti-communities, are explored, tested, and adapted with a wide range of networks from disparate fields including neuroscience, ecology, geology, social ecology, linguistics, network theory, psychology, microbiology, gene sequencing, business corporations, and sociology. In particular, to better approximate typical structural features of real-world networks, Erd˝os–R´enyi and scale-free network random models are modified by adding select subgraphs to match real-world networks with the aim of improving their usefulness in network analysis. Before looking for anti-communities we first look at ways to partition graphs into communities. As well as generic techniques such as local improvement methods and spectral partitioning, a number of specialised methods are studied. These include link centrality, similarity, communicability, optimisation, and the Louvain method. We then look at topics associated with near bipartivity in real world graphs. Particular attention is given to the measurement of bipartivity and anti-communities, and their definitions are broadened, refined and tested so that many more networks that demonstrate varying degrees of bipartivity can be analysed using those methods. We prove new theoretical results showing how widely measures can differ and then look to determine the best measures to use, as well as the most effective structure-revealing algorithms. Thorough testing involves nearly one hundred real-world networks including some which have a known tendency for bipartivity (such as airlines networks, and fullerene graphs) as well as near-bipartite graphs based on random trees (including those generated from Pr¨ufer sequences, and other artificially constructed examples. We are able to give conclusions about the measures and methods that should be employed in practice.As real-world networks grow increasingly complex, new and adaptable methods are required to analyse and understand the defining features of these networks. To find and exploit these unique network features, current methods, primarily for random model generation and detection of anti-communities, are explored, tested, and adapted with a wide range of networks from disparate fields including neuroscience, ecology, geology, social ecology, linguistics, network theory, psychology, microbiology, gene sequencing, business corporations, and sociology. In particular, to better approximate typical structural features of real-world networks, Erd˝os–R´enyi and scale-free network random models are modified by adding select subgraphs to match real-world networks with the aim of improving their usefulness in network analysis. Before looking for anti-communities we first look at ways to partition graphs into communities. As well as generic techniques such as local improvement methods and spectral partitioning, a number of specialised methods are studied. These include link centrality, similarity, communicability, optimisation, and the Louvain method. We then look at topics associated with near bipartivity in real world graphs. Particular attention is given to the measurement of bipartivity and anti-communities, and their definitions are broadened, refined and tested so that many more networks that demonstrate varying degrees of bipartivity can be analysed using those methods. We prove new theoretical results showing how widely measures can differ and then look to determine the best measures to use, as well as the most effective structure-revealing algorithms. Thorough testing involves nearly one hundred real-world networks including some which have a known tendency for bipartivity (such as airlines networks, and fullerene graphs) as well as near-bipartite graphs based on random trees (including those generated from Pr¨ufer sequences, and other artificially constructed examples. We are able to give conclusions about the measures and methods that should be employed in practice

Computing Shrub-Depth Decompositions

Shrub-depth is a width measure of graphs which, roughly speaking, corresponds to the smallest depth of a tree into which a graph can be encoded. It can be thought of as a low-depth variant of clique-width (or rank-width), similarly as treedepth is a low-depth variant of treewidth. We present an fpt algorithm for computing decompositions of graphs of bounded shrub-depth. To the best of our knowledge, this is the first algorithm which computes the decomposition directly, without use of rank-width decompositions and FO or MSO logic