290 research outputs found
The Expressive Power of CSP-Quantifiers
A generalized quantifier QK is called a CSP-quantifier if its defining class K consists of all structures that can be homomorphically mapped to a fixed finite template structure. For all positive integers n ≥ 2 and k, we define a pebble game that characterizes equivalence of structures with respect to the logic Lk∞ω(CSP+n ), where CSP+n is the union of the class Q1 of all unary quantifiers and the class CSPn of all CSP-quantifiers with template structures that have at most n elements. Using these games we prove that for every n ≥ 2 there exists a CSP-quantifier with template of size n + 1 which is not definable in Lω∞ω(CSP+n ). The proof of this result is based on a new variation of the well-known Cai-Fürer-Immerman construction.publishedVersionPeer reviewe
The long-range Falicov-Kimball model and the amorphous Kitaev model: Quantum many-body systems I have known and loved
Large systems of interacting objects can give rise to a rich array of emergent behaviours. Make those objects quantum and the possibilities only expand. Interacting quantum many-body systems, as such systems are called, include essentially all physical systems. Luckily, we don't usually need to consider this full quantum many-body description. The world at the human scale is essentially classical (not quantum), while at the microscopic scale of condensed matter physics we can often get by without interactions. Strongly correlated materials, however, do require the full description. Some of the most exciting topics in modern condensed matter fall under this umbrella: the spin liquids, the fractional quantum Hall effect, high temperature superconductivity and much more. Unfortunately, strongly correlated materials are notoriously difficult to study, defying many of the established theoretical techniques within the field. Enter exactly solvable models, these are interacting quantum many-body systems with extensively many local symmetries. The symmetries give rise to conserved charges. These charges break the model up into many non-interacting quantum systems which are more amenable to standard theoretical techniques. This thesis will focus on two such exactly solvable models.
The first, the Falicov-Kimball (FK) model is an exactly solvable limit of the famous Hubbard model which describes itinerant fermions interacting with a classical Ising background field. Originally introduced to explain metal-insulator transitions, it has a rich set of ground state and thermodynamic phases. Disorder or interactions can turn metals into insulators and the FK model features both transitions. We will define a generalised FK model in 1D with long-range interactions. This model shows a similarly rich phase diagram to its higher dimensional cousins. We use an exact Markov Chain Monte Carlo method to map the phase diagram and compute the energy resolved localisation properties of the fermions. This allows us to look at how the move to 1D affects the physics of the model. We show that the model can be understood by comparison to a simpler model of fermions coupled to binary disorder.
The second, the Kitaev Honeycomb (KH) model, was the one of the first solvable 2D models with a Quantum Spin Liquid (QSL) ground state. QSLs are generally expected to arise from Mott insulators, when frustration prevents magnetic ordering all the way to zero temperature. The QSL state defies the traditional Landau-Ginzburg-Wilson paradigm of phases being defined by local order parameters. It is instead a topologically ordered phase. Recent work generalising non-interacting topological insulator phases to amorphous lattices raises the question of whether interacting phases like the QSLs can be similarly generalised. We extend the KH model to random lattices with fixed coordination number three generated by Voronoi partitions of the plane. We show that this model remains solvable and hosts a chiral amorphous QSL ground state. The presence of plaquettes with an odd number of sides leads to a spontaneous breaking of time reversal symmetry. We unearth a rich phase diagram displaying Abelian as well as a non-Abelian QSL phases with a remarkably simple ground state flux pattern. Furthermore, we show that the system undergoes a phase transition to a conducting thermal metal state and discuss possible experimental realisations.Open Acces
A guide to the Rado graph : exploring structural and logical properties of the Rado graph
Dissertation (MSc (Mathematics))--University of Pretoria, 2023.The Rado graph, denoted R, is the unique (up to isomorphism) countably infinite random graph. It satisfies the extension property, that is, for two finite sets U and V of vertices of R there is a vertex outside of both U and V connected to every vertex in U and none in V . This property of the Rado graph allows us to prove
quite a number of interesting results, such as a 0-1-law for graphs. Amongst other things, the Rado graph is partition regular, non-fractal, ultrahomogeneous, saturated, resplendent, the Fra´ıss´e-limit of the class of finite graphs, a non-standard model of the first-order theory of finite graphs, and has a complete decidable theory.
We classify the definable subgraphs of the Rado graph and prove results for finite graphs that satisfy a restricted version of the extension property. We also mention some parallels between the rationals viewed as a linear order and the Rado graph.Mathematics and Applied MathematicsMSc (Mathematics)Unrestricte
On learning the structure of clusters in graphs
Graph clustering is a fundamental problem in unsupervised learning, with numerous applications in computer science and in analysing real-world data. In many real-world applications, we find that the clusters have a significant high-level structure. This is often overlooked in the design and analysis of graph clustering algorithms which make strong simplifying assumptions about the structure of the graph. This thesis addresses the natural question of whether the structure of clusters can be learned efficiently and describes four new algorithmic results for learning such structure in graphs and hypergraphs.
The first part of the thesis studies the classical spectral clustering algorithm, and presents a tighter analysis on its performance. This result explains why it works under a much weaker and more natural condition than the ones studied in the literature, and helps to close the gap between the theoretical guarantees of the spectral clustering algorithm and its excellent empirical performance.
The second part of the thesis builds on the theoretical guarantees of the previous part and shows that, when the clusters of the underlying graph have certain structures, spectral clustering with fewer than k eigenvectors is able to produce better output than classical spectral clustering in which k eigenvectors are employed, where k is the number of clusters. This presents the first work that discusses and analyses the performance of spectral clustering with fewer than k eigenvectors, and shows that general structures of clusters can be learned with spectral methods.
The third part of the thesis considers efficient learning of the structure of clusters with local algorithms, whose runtime depends only on the size of the target clusters and is independent of the underlying input graph. While the objective of classical local clustering algorithms is to find a cluster which is sparsely connected to the rest of the graph, this part of the thesis presents a local algorithm that finds a pair of clusters which are densely connected to each other. This result demonstrates that certain structures of clusters can be learned efficiently in the local setting, even in the massive graphs which are ubiquitous in real-world applications.
The final part of the thesis studies the problem of learning densely connected clusters in hypergraphs. The developed algorithm is based on a new heat diffusion process, whose analysis extends a sequence of recent work on the spectral theory of hypergraphs. It allows the structure of clusters to be learned in datasets modelling higher-order relations of objects and can be applied to efficiently analyse many complex datasets occurring in practice.
All of the presented theoretical results are further extensively evaluated on both synthetic and real-word datasets of different domains, including image classification and segmentation, migration networks, co-authorship networks, and natural language processing. These experimental results demonstrate that the newly developed algorithms are practical, effective, and immediately applicable for learning the structure of clusters in real-world data
Subchromatic numbers of powers of graphs with excluded minors
A -subcolouring of a graph is a function
such that the set of vertices coloured induce a disjoint union of cliques.
The subchromatic number, , is the minimum such that
admits a -subcolouring. Ne\v{s}et\v{r}il, Ossona de Mendez, Pilipczuk,
and Zhu (2020), recently raised the problem of finding tight upper bounds for
when is planar. We show that
when is planar, improving their bound of
135. We give even better bounds when the planar graph has larger girth.
Moreover, we show that , improving the
previous bound of 364. For these we adapt some recent techniques of Almulhim
and Kierstead (2022), while also extending the decompositions of triangulated
planar graphs of Van den Heuvel, Ossona de Mendez, Quiroz, Rabinovich and
Siebertz (2017), to planar graphs of arbitrary girth. Note that these
decompositions are the precursors of the graph product structure theorem of
planar graphs.
We give improved bounds for for all , whenever
has bounded treewidth, bounded simple treewidth, bounded genus, or excludes
a clique or biclique as a minor. For this we introduce a family of parameters
which form a gradation between the strong and the weak colouring numbers. We
give upper bounds for these parameters for graphs coming from such classes.
Finally, we give a 2-approximation algorithm for the subchromatic number of
graphs coming from any fixed class with bounded layered cliquewidth. In
particular, this implies a 2-approximation algorithm for the subchromatic
number of powers of graphs coming from any fixed class with bounded
layered treewidth (such as the class of planar graphs). This algorithm works
even if the power and the graph is unknown.Comment: 21 pages, 2 figure
On the Quantum Chromatic Numbers of Small Graphs
We make two contributions pertaining to the study of the quantum chromatic
numbers of small graphs. Firstly, in an elegant paper, Man\v{c}inska and
Roberson [\textit{Baltic Journal on Modern Computing}, 4(4), 846-859, 2016]
gave an example of a graph on 14 vertices with quantum chromatic
number 4 and classical chromatic number 5, and conjectured that this is the
smallest graph exhibiting a separation between the two parameters. We describe
a computer-assisted proof of this conjecture, thereby resolving a longstanding
open problem in quantum graph theory. Our second contribution pertains to the
study of the rank- quantum chromatic numbers. While it can now be shown that
for every , and are distinct, few small examples of
separations between these parameters are known. We give the smallest known
example of such a separation in the form of a graph on 21 vertices
with and . The previous record was held by a
graph on 57 vertices that was first considered in the aforementioned
paper of Man\v{c}inska and Roberson and which satisfies
and . In addition, provides the first
provable separation between the parameters and .
We believe that our techniques for constructing and lower bounding its
orthogonal rank could be of independent interest
On graphs with no induced or
In this paper, we are interested in some problems related to chromatic number
and clique number for the class of -free graphs, and prove the
following. If is a connected ()-free graph with
, then either is the complement of a bipartite graph or
has a clique cut-set. Moreover, there is a connected ()-free
imperfect graph with and has no clique cut-set. This
strengthens a result of Malyshev and Lobanova [Disc. Appl. Math. 219 (2017)
158--166]. If is a ()-free graph with ,
then . Moreover, the bound is tight when
. This result together with known results partially
answers a question of Ju and Huang [arXiv:2303.18003 [math.CO] 2023], and also
improves a result of Xu [Manuscript 2022].
While the "Chromatic Number Problem" is known to be -hard for the class
of -free graphs, our results together with some known results imply that
the "Chromatic Number Problem" can be solved in polynomial time for the class
of ()-free graphs which may be independent interest.Comment: This paper is dedicated to the memory of Professor Frederic Maffray
on his death anniversar
Near Optimal Colourability on Hereditary Graph Families
A graph family is near optimal colourable if there is a
constant number such that every graph satisfies
, where and are the
chromatic number and clique number of , respectively. The near optimal
colourable graph families together with the Lov{\'a}sz theta function are
useful for the study of the chromatic number problems for hereditary graph
families. In this paper, we investigate the near optimal colourability for
()-free graphs. Our main result is an almost complete characterization
for the near optimal colourability for ()-free graphs with two
exceptional cases, one of which is the celebrated Gy{\'a}rf{\'a}s conjecture.
To obtain the result, we prove that the family of ()-free
graphs is near optimal colourable for every positive integer by inductive
arguments.Comment: 11 pages, 1 figur
Hardness Transitions of Star Colouring and Restricted Star Colouring
We study how the complexity of the graph colouring problems star colouring
and restricted star colouring vary with the maximum degree of the graph.
Restricted star colouring (in short, rs colouring) is a variant of star
colouring. For , a -colouring of a graph is a function
such that for every edge of
. A -colouring of is called a -star colouring of if there is
no path in with and . A -colouring of
is called a -rs colouring of if there is no path in with
. For , the problem -STAR COLOURABILITY
takes a graph as input and asks whether admits a -star colouring.
The problem -RS COLOURABILITY is defined similarly. Recently, Brause et al.
(Electron. J. Comb., 2022) investigated the complexity of 3-star colouring with
respect to the graph diameter. We study the complexity of -star colouring
and -rs colouring with respect to the maximum degree for all . For
, let us denote the least integer such that -STAR COLOURABILITY
(resp. -RS COLOURABILITY) is NP-complete for graphs of maximum degree by
(resp. ).
We prove that for and , -STAR COLOURABILITY is NP-complete
for graphs of maximum degree . We also show that -RS COLOURABILITY is
NP-complete for planar 3-regular graphs of girth 5 and -RS COLOURABILITY is
NP-complete for triangle-free graphs of maximum degree for .
Using these results, we prove the following: (i) for and ,
-STAR COLOURABILITY is NP-complete for -regular graphs if and only if
; and (ii) for , -RS COLOURABILITY is NP-complete
for -regular graphs if and only if
Coloring polygon visibility graphs and their generalizations
Curve pseudo-visibility graphs generalize polygon and pseudo-
polygon visibility graphs and form a hereditary class of
graphs. We prove that every curve pseudo-visibility graph
with clique number ω has chromatic number at most 3 · 4ω−1.
The proof is carried through in the setting of ordered graphs;
we identify two conditions satisfied by every curve pseudo-
visibility graph (considered as an ordered graph) and prove
that they are sufficient for the claimed bound. The proof is
algorithmic: both the clique number and a coloring with the
claimed number of colors can be computed in polynomial time
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