Combinatorics and Optimization

Permanent URI for this collectionhttps://uwspace.uwaterloo.ca/handle/10012/9928

This is the collection for the University of Waterloo's Department of Combinatorics and Optimization.

Research outputs are organized by type (eg. Master Thesis, Article, Conference Paper).

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Now showing 1 - 13 of 13

Algebraic Approach to Quantum Isomorphisms
(University of Waterloo, 2024-09-24) Sobchuk, Mariia; Godsil, Chris
In very brief, this thesis is a study of quantum isomorphisms. We have started with two pairs of quantum isomorphic graphs and looked for generalizations of those. We have learned that those two pairs of graphs are related by Godsil-McKay switching and one of the graphs is an orthogonality graph of lines in a root system. These two observations lead to research in two directions. First, since quantum isomorphisms preserve coherent algebras, we studied a question of when Godsil-McKay switching preserved coherent algebras. In this way, non isomorphic graphs related by Godsil-McKay switching with isomorphic coherent algebras are candidates to being quantum isomorphic and non isomorphic. Second, while it was known that one of the graphs in a pair was an orthogonality graphs of the lines in a root system $E_8,$ we showed that a graph from another pair is also an orthogonality graph of the lines in a root system $F_4.$ We have studied orthogonality graphs of lines in root systems $B_{2^d},C_{2^d},D_{2^d}$ and showed that they have quantum symmetry. Finally, we have touched upon structures of quantum permutations, relationships between fractional and quantum isomorphisms as well as connection to quantum independence and chromatic numbers.
Bipartite Quantum Walks and the Hamiltonian
(University of Waterloo, 2023-09-26) Chen, Qiuting; Godsil, Chris
We study a discrete quantum walk model called bipartite walks via a spectral approach. A bipartite walk is determined by a unitary matrix U, i.e., the transition matrix of the walk. For every transition matrix U, there is a Hamiltonian H such that U = exp(iH). If there is a real skew-symmetric matrix S such that H = iS, we say there is a H-digraph associated to the walk and S is the skew-adjacency matrix of the H-digraph. The underlying unweighted non-oriented graph of the H-digraph is H-graph. Let G be a simple bipartite graph with no isolated vertices. The bipartite walk on G is the same as the continuous walk on the H-digraph over integer time. Two questions lie in the centre of this thesis are 1. Is there a connection between the H-(di)graph and the underlying graph G? If there is, what is the connection? 2. Is there a connection between the walk and the underlying graph G? If there is, what is the connection? Given a bipartite walk on G, we show that the underlying bipartite graph G determines the existence of the H-graph. If G is biregular, the spectrum of G determines the spectrum of U. We give complete characterizations of bipartite walks on paths and even cycles. Given a path or an even cycle, the transition matrix of the bipartite walk is a permutation matrix. The H-digraph is an oriented weighted complete graph. Using bipartite walks on even paths, we construct a in nite family of oriented weighted complete graphs such that continuous walks de- ned on them have universal perfect state transfer, which is an interesting but rare phenomenon. Grover's walk is one of the most studied discrete quantum walk model and it can be used to implement the famous Grover's algorithm. We show that Grover's walk is actually a special case of bipartite walks. Moreover, given a bipartite graph G, one step of the bipartite walk on G is the same as two steps of Grover's walk on the same graph. We also study periodic bipartite walks. Using results from algebraic number theory, we give a characterization of periodic walks on a biregular graph with a constraint on its spectrum. This characterization only depends on the spectrum of the underlying graph and the possible spectrum for a periodic walk is determined by the degrees of the underlying graph. We apply this characterization of periodic bipartite walk to Grover's walk to get a characterization of a certain class of periodic Grover's walk. Lastly, we look into bipartite walks on the incidence graphs of incidence structures, t-designs (t 2) and generalized quadrangles in particular. Given a bipartite walk on a t-design, we show that if the underlying design is a partial linear space, the H-graph is the distance-two graph of the line graph of the underlying incidence graph. Given a bipartite walk on the incidence graph of a generalized quadrangle, we show that there is a homogeneous coherent algebra raised from the bipartite walk. This homogeneous coherent algebra is useful in analyzing the behavior of the walk.
Constructing Cospectral and Comatching Graphs
(University of Waterloo, 2019-07-18) Wang, Xiaojing; Godsil, Chris
The matching polynomial is a graph polynomial that does not only have interesting mathematical properties, but also possesses meaningful applications in physics and chemistry. For a simple graph, the matching polynomial enumerates the number of matchings of different sizes in it. Two graphs are comatching if they have the same matching polynomial. Two vertices u, v in a graph G are comatching if G\ u and G\ v are comatching. In 1973, Schwenk proved almost every tree has the same characteristic polynomial with another tree. In this thesis, we extend Schwenk's result to maximal limbs and weighted trees. We also give a construction using 1-vertex extensions for comatching graphs and graphs with an arbitrarily large number of comatching vertices. In addition, we use an alternative definition of matching polynomial for multigraphs to derive new identities for the matching polynomial. These identities are tools used towards our 2-sum construction for comatching vertices and comatching graphs.
Covering Graphs and Equiangular Tight Frames
(University of Waterloo, 2016-09-02) Rahimi, Fahimeh; Godsil, Chris
Recently, there has been huge attention paid to equiangular tight frames and their constructions, due to the fact that the relationship between these frames and quantum information theory was established. One of the problems which has been studied is the relationship between equiangular tight frames and covering graphs of complete graphs. In this thesis, we will explain equiangular tight frames and covering graphs of complete graphs and present the results that show the relationship between these two concepts. The latest results about the constructions of equiangular tight frames from projective geometries and Steiner systems also has been presented.
Discrete Quantum Walks on Graphs and Digraphs
(University of Waterloo, 2018-09-26) Zhan, Hanmeng; Godsil, Chris
This thesis studies various models of discrete quantum walks on graphs and digraphs via a spectral approach. A discrete quantum walk on a digraph $X$ is determined by a unitary matrix $U$, which acts on complex functions of the arcs of $X$. Generally speaking, $U$ is a product of two sparse unitary matrices, based on two direct-sum decompositions of the state space. Our goal is to relate properties of the walk to properties of $X$, given some of these decompositions. We start by exploring two models that involve coin operators, one due to Kendon, and the other due to Aharonov, Ambainis, Kempe, and Vazirani. While $U$ is not defined as a function in the adjacency matrix of the graph $X$, we find exact spectral correspondence between $U$ and $X$. This leads to characterization of rare phenomena, such as perfect state transfer and uniform average vertex mixing, in terms of the eigenvalues and eigenvectors of $X$. We also construct infinite families of graphs and digraphs that admit the aforementioned phenomena. The second part of this thesis analyzes abstract quantum walks, with no extra assumption on $U$. We show that knowing the spectral decomposition of $U$ leads to better understanding of the time-averaged limit of the probability distribution. In particular, we derive three upper bounds on the mixing time, and characterize different forms of uniform limiting distribution, using the spectral information of $U$. Finally, we construct a new model of discrete quantum walks from orientable embeddings of graphs. We show that the behavior of this walk largely depends on the vertex-face incidence structure. Circular embeddings of regular graphs for which $U$ has few eigenvalues are characterized. For instance, if $U$ has exactly three eigenvalues, then the vertex-face incidence structure is a symmetric $2$-design, and $U$ is the exponential of a scalar multiple of the skew-symmetric adjacency matrix of an oriented graph. We prove that, for every regular embedding of a complete graph, $U$ is the transition matrix of a continuous quantum walk on an oriented graph.
Distance-Biregular Graphs and Orthogonal Polynomials
(University of Waterloo, 2023-09-15) Lato, Sabrina; Godsil, Chris
This thesis is about distance-biregular graphs– when they exist, what algebraic and structural properties they have, and how they arise in extremal problems. We develop a set of necessary conditions for a distance-biregular graph to exist. Using these conditions and a computer, we develop tables of possible parameter sets for distancebiregular graphs. We extend results of Fiol, Garriga, and Yebra characterizing distance-regular graphs to characterizations of distance-biregular graphs, and highlight some new results using these characterizations. We also extend the spectral Moore bounds of Cioaba et al. to semiregular bipartite graphs, and show that distance-biregular graphs arise as extremal examples of graphs meeting the spectral Moore bound.
Edge State Transfer
(University of Waterloo, 2019-01-11) Chen, Qiuting; Godsil, Chris
Let G be a graph and let t be a positive real number. Then the evolution of the continuous quantum walk defined on G is described by the transition matrix U(t)=exp(itH).The matrix H is called Hamiltonian. So far the most studied quantum walks are the ones whose Hamiltonians are the adjacency matrices of the underlying graphs and initial states are vertex states e_a, with e_a being the characteristic vector of vertex a. This thesis focuses on Laplacian edge state transfer, that is, the quantum walks whose initial states are edge states e_a-e_b and Hamiltonians are the Laplacians of the underlying graphs. So far the research about perfect state transfer only involves vertex states and Laplacian edge state transfer has not been studied before. We extend the known results of perfect vertex state transfer and periodicity of vertex states to Laplacian edge state transfer. We prove two useful closure properties for perfect Laplacian edge state transfer. One is that complementation preserves perfect edge state transfer. The other is that if G has perfect Laplacian edge state transfer at time τ and H has perfect Laplacian vertex state transfer also at time τ, then with some mild assumption on the pairs of vertex states and edge states that have perfect state transfer, the Cartesian product G □ H also admits perfect edge state transfer. Those two properties provide us new ways to construct graphs with perfect Laplacian edge state transfer. We also observe one phenomenon that happens in Laplacian edge state transfer which never happens in vertex state transfer: if there is perfect state transfer from e_a-e_b to e_α-e_β and also from e_b-e_c to e_β-e_γ at the same time t in G, then G admits perfect state transfer from e_a-e_c to e_α-e_γ at time t. We give characterizations of perfect Laplacian edge state transfer in cycles, paths and complete bipartite graphs K_{2,4n}. We study perfect state transfer and periodicity on edge states with special spectral features. We also consider the case when the unsigned Laplacian is Hamiltonian and initial states are plus states of the form e_a+e_b. In this case, we characterize perfect state transfer in paths, cycles and bipartite graphs. We close this thesis by a list of open questions.
Hurwitz Trees and Tropical Geometry
(University of Waterloo, 2016-01-21) Akeyr, Garnet Jonathan; McKinnon, David; Godsil, Chris
The lifting problem in algebraic geometry asks when a finite group G acting on a curve defined over characteristic p > 0 lifts to characteristic 0. One object used in the study of this problem is the Hurwitz tree, which encodes the ramification data of a group action on a disk. In this thesis we explore the connection between Hurwitz trees and tropical geometry. That is, we can view the Hurwitz tree as a tropical curve. After exploring this connection we provide two examples to illustrate the connection, using objects in tropical geometry to demonstrate when a group action fails to lift.
Matchings and Representation Theory
(University of Waterloo, 2018-12-20) Lindzey, Nathan; Cheriyan, Joseph; Godsil, Chris
In this thesis we investigate the algebraic properties of matchings via representation theory. We identify three scenarios in different areas of combinatorial mathematics where the algebraic structure of matchings gives keen insight into the combinatorial problem at hand. In particular, we prove tight conditional lower bounds on the computational complexity of counting Hamiltonian cycles, resolve an asymptotic version of a conjecture of Godsil and Meagher in Erdos-Ko-Rado combinatorics, and shed light on the algebraic structure of symmetric semidefinite relaxations of the perfect matching problem
Quantum independence and chromatic numbers
(University of Waterloo, 2019-08-28) Sobchuk, Mariia; Godsil, Chris
In this thesis we are studying the cases when quantum independence and quantum chromatic numbers coincide with or differ from their classical counterparts. Knowing about the relation of chromatic numbers separation to the projective Kochen-Specker sets, we found an analogous characterisation for the independence numbers case. Additionally, all the graphs that we studied that had known quantum parameters exhibited both the separation between the classical and quantum independence numbers and the separation between the classical and quantum chromatic numbers. This observation and the Kochen-Specker connection suggested the possibility of the chromatic and independence numbers separations occurring simultaneously. We have disproved this idea with a counterexample. Furthermore, we generalised Manĉinska-Roberson’s example of the chromatic numbers separation to an infinite family. We investigate some known instances with strictly larger quantum independence numbers in-depth, find a more general description and generalise Piovesan’s example. Using the Lovász theta bound, we prove that there is no separation between the independence numbers in bipartite and perfect graphs. We also show that there is no separation when the classical independence number is two; and that the cone over a graph has the same quantum independence number as the underlying graph.
Quantum Walks and Pretty Good State transfer on Paths
(University of Waterloo, 2019-08-23) van Bommel, Christopher Martin; Godsil, Chris
Quantum computing is believed to provide many advantages over traditional computing, particularly considering the speed at which computations can be performed. One of the challenges that needs to be resolved in order to construct a quantum computer is the transmission of information from one part of the computer to another. Quantum walks, the quantum analogues of classical random walks, provide one potential method for resolving this challenge. In this thesis, we use techniques from algebraic graph theory and number theory to analyze the mathematical model for continuous time quantum walks on graphs. For the continuous time quantum walk model, we define a transition operator, which is a function of a Hamiltonian. We focus on the cases where the adjacency matrix or the Laplacian of a graph act as the Hamiltionian. We mainly consider quantum walks on paths as a model for spin chains, which are the underlying basis of a quantum communication protocol. For communication to be efficient, we desire states to be transferred with high fidelity, a measure of the amount of similarity between the transmitted state and the received state. At the maximum fidelity of 1, we say we have achieved perfect state transfer. Examples of perfect state transfer are relatively rare, so the concept of pretty good state transfer was introduced as a natural relaxation, which exists if fidelities arbitrarily close to 1 are obtained. Our first main result is to characterize pretty good state transfer on paths. Previously, pretty good state transfer on paths was considered mainly for the end vertices, though results for both models indicated that if there was pretty good state transfer between the end vertices, then there was pretty good state transfer between internal vertices equidistant from each end. We complete the characterization by demonstrating, for the adjacency matrix model, a family of paths where pretty good state transfer exists between internal vertices but not between end vertices, and verifying that no other example exists. For the Laplacian model, we show that there are no paths with pretty good state transfer between internal vertices but not between the end vertices. Our second main result considers initial states involving multiple vertices. Under the adjacency matrix model, we provide necessary and sufficient conditions for pretty good state transfer in a particular family of paths in terms of the eigenvalue support of the initial state. We also discuss recent results on fractional revival, which is another form of multiple qubit state transfer.
Quantum Walks on Oriented Graphs
(University of Waterloo, 2019-01-11) Lato, Sabrina; Godsil, Chris
This thesis extends results about periodicity and perfect state transfer to oriented graphs. We prove that if a vertex a is periodic, then elements of the eigenvalue support lie in Z √∆ for some squarefree negative integer ∆. We find an infinite family of orientations of the complete graph that are periodic. We find an example of a graph with both perfect state transfer and periodicity that is not periodic at an integer multiple of the period, and we prove and use Gelfond-Schneider Theorem to show that every oriented graph with perfect state transfer between two vertices will have both vertices periodic. We find a complete characterization of when perfect state transfer can occur in oriented graphs, and find a new example of a graph where one vertex has perfect state transfer to multiple other vertices.
State Transfer & Strong Cospectrality in Cayley Graphs
(University of Waterloo, 2022-08-09) Árnadóttir, Arnbjörg Soffía; Godsil, Chris
This thesis is a study of two graph properties that arise from quantum walks: strong cospectrality of vertices and perfect state transfer. We prove various results about these properties in Cayley graphs. We consider how big a set of pairwise strongly cospectral vertices can be in a graph. We prove an upper bound on the size of such a set in normal Cayley graphs in terms of the multiplicities of the eigenvalues of the graph. We then use this to prove an explicit bound in cubelike graphs and more generally, Cayley graphs of $Z_2^{d_1} \times Z_4^{d_2}$. We further provide an infinite family of examples of cubelike graphs (Cayley graphs of $Z_2^d$ ) in which this set has size at least four, covering all possible values of $d$. We then look at perfect state transfer in Cayley graphs of abelian groups having a cyclic Sylow-2-subgroup. Given such a group, G, we provide a complete characterization of connection sets C such that the corresponding Cayley graph for G admits perfect state transfer. This is a generalization of a theorem of Ba\v{s}i\'{c} from 2013, where he proved a similar characterization for Cayley graphs of cyclic groups.

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