This thesis deals with the applications of graph theory towards the electrical distribution networks that transmit electricity from the generators that produce it and the consumers that use it. Specifically, we establish the substation and bus network as graph theoretical models for this major piece of electrical infrastructure. We also generate substation and bus networks for a wide range of existing data from both synthetic and real grids and show several properties of these graphs, such as density, degeneracy, and planarity. We also motivate future research into the definition of a graph family containing bus and substation networks and the classification of that family as having polynomial expansion.

Mathematical models of disease spreading are a key factor of ensuring that we are prepared to deal with the next epidemic. They allow us to predict how an infection will spread throughout a population, thereby allowing us to make intelligent choices when attempting to contain the disease. Whether due to a lack of empirical data, a lack of computational power, a lack of biological understanding, or some combination thereof, traditional models must make sweeping assumptions about the behavior of a population during an epidemic. In this thesis, we implement granular epidemic simulations using a rich social network constructed from real-world interactions. We develop computational models for three diseases, and we use these simulations to demonstrate the effects of twelve potential intervention strategies, both before and during a hypothetical epidemic. We show how representing a population as a temporal graph and applying existing graph metrics can lead to more effective interventions.

Rigidity theory deals in problems of the following form: given a structure defined by geometric constraints on a set of objects, what information about its geometric behavior is implied by the underlying combinatorial structure. The most well-studied class of structures is the bar-joint framework, which is made of fixed-length bars connected by universal joints with full rotational degrees of freedom; the allowed motions preserve the lengths and connectivity of the bars, and a framework is rigid if the only allowed motions are trivial motions of Euclidean space. A remarkable theorem of Maxwell-Laman says that rigidity of generic bar-joint frameworks depends only on the graph that has as its edges the bars and as its vertices the joints. We generalize the "degree of freedom counts that appear in the Maxwell-Laman theorem to the very general setting of (k,l)-sparse and (k,l)-graded sparse hypergraphs. We characterize these in terms of their graph-graph theoretic and matroidal properties. For the fundamental algorithmic problems Decision, Extraction, Components, and Decomposition, we give efficient, implementable pebble game algorithms for all the (k,l)-sparse and (k,l)-graded-sparse families of hypergraphs we study. We then prove that all the matroids arising from (k,l)-sparse are linearly representable by matrices with a certain "natural" structure that captures the incidence structure of the hypergraph and the sparsity parameters k and l. Building on the combinatorial and linear theory discussed above, we introduce a new rigidity model: slider-pinning rigidity. This is an elaboration of the planar bar-joint model to include sliders, which constrain a vertex to move on a specific line. We prove the analogue of the Maxwell-Laman Theorem for slider pinning, using, as a lemma, a new proof of Whiteley's Parallel Redrawing Theorem. We conclude by studying the emergence of non-trivial rigid substructures in generic planar frameworks given by Erdos-Renyi random graphs. We prove that there is a sharp threshold for such substructures to emerge, and that, when they do, they are all linear size. This is consistent with experimental and simulation-based work done in the physics community on the formation of certain glasses.

Ontology-based technologies like Semantic Web and Knowledge Graphs are promising for knowledge management in manufacturing industries. In the literature there are abundant of publications related to using ontologies to represent and capture knowledge in manufacturing. Many of them cover the use of ontologies for managing knowledge in different aspects of Product Lifecycle Management (PLM). Nevertheless, very few of them cover how ontologies can be used with virtual factory models, data and information as well as the knowledge generated from using these models and their corresponding engineering activities. An “extension” of existing ontologies is badly needed as digital, virtual models in terms of simulation and digital twins have become more popular in the industry. Without such an extended knowledge management process and system, it is difficult to re-use the artefacts and knowledge generated from the expensive and valuable virtual engineering activities. Relying on the cutting-edge graph database technologies and what they can offer regarding knowledge management, and also recent developments in the domain ontology field, an extended knowledge management implementation, specifically designed for virtual engineering has been done. Moreover, a clear roadmap for establishment of knowledge bases around production systems armed with Virtual Factory(VF) and Multi-Objective Optimization (MOO) processes has been provided. This, includes defining key elements of manufacturing procedures, constructing an ontology, defining data structure in preferably a graph database, and accessing valuable historical (provenance) data regarding different engineering entities and/or activities.

The aim of this thesis was to study co-authorship in the constraint programming research community. This was done by conducting social network analysis (SNA) based on published scientific papers from the proceedings of the International Conference on Principles and Practice of Constraint Programming. Bibliographic data of the scientific literature was collected for the years 2018–2022 of the annual conference. For quantitative analysis, graph metrics were computed to study the properties and structure of the overall network, and also to study the attributes and characteristics of individual authors to be able to identify central actors of the community. Furthermore, graph layout algorithms were used for visualisation of the network. The computed metrics and the graphical visualisations enabled identifying collaboration patterns and behaviours within the studied field. The results of this study show that the most central actors of the community are mainly male and dominated by white organisations and countries. The results of the study also show that the vast majority of authors of the community collaborate with others in writing papers. However, due to the low density of the network there is opportunity and room for new collaboration patterns to take place within the research community.

<p dir="ltr">Chemical reaction networks govern polymer degradation and contain critical design information regarding specific susceptibilities, degradation pathways, and degradants. However, predicting reaction pathways and characterizing complete reaction networks has been hindered by high computational costs because of the vast number of possible reactions at deeper levels of network exploration. In the first section, an exploration policy based on Dijkstra's algorithm on YARP using the reaction rate as a cost function was shown to provide a tractable means of exploring the pyrolytic degradation network of a representative commodity polymer, PEG. The resulting network is the largest reported to date for this system and includes pathways out to all degradants observed in earlier mass spectrometry studies. The initial degradation pathway predictions were validated by complementary experimental analysis of pyrolyzed PEG samples by ESI-MS. These findings demonstrate that reaction network characterization is reaching sufficient maturity to be used as an exploratory tool for investigating materials degradation and interpreting experimental degradation studies.</p>

We present a theoretical study that investigates the applicability of a graph theoretical approach to realize various quantum experiments. Crucially, we may represent quantum optical experiments involving tabletop optical elements in terms of highly interpretable, coloured, weighted multi-graphs. We introduce the formalism behind this approach; then through the digital discovery framework PyTheus, we uncover over 100 different quantum experiments which realizes complex, novel quantum states. Towards enhancing our interpretation of the AI-based framework's solutions, we also leverage eXplainable-AI (XAI) techniques from computer vision to investigate what a trained neural network learns about quantum experiments. Crucially, we find that we are able to conceptualize the learned strategies which the neural network applies to optimize for a target quantum property, and discover how the network conceives of its solution. We conclude by presenting an experimental proposal which yields realizable solutions that, for the first time, solves high-dimensional variants of a quantum retrodiction puzzle known as the Mean King's Problem. We, therefore, present a case study which investigates the potential for new scientific discoveries through a joint collaboration between human and artificial intelligence.

Electron-driven processes lie at the core of a large variety of physical, biological, and chemical phenomena. Despite their crucial roles in science and technology, detailed description of these processes remains a significant challenge, and there is a need for the development of accurate and efficient computational tools that enable predictive simulation. This work is focused on the development of novel software tools and methodologies aimed at two classes of electron-mediated processes: (i) electron-molecule scattering, and (ii) charge transfer in proteins. The first major focus of this thesis is the electronic structure of autoionizing electronic resonances. The theoretical description of these metastable states is intractable by means of conventional quantum chemistry techniques, and specialized techniques are required in order to accurately describe their energies and lifetimes. In this work, we have utilized the complex absorbing potential (CAP) method, and describe three developments which have advanced the applicability, efficiency, and accessibility of the CAP methodology for molecular resonances: (1) implementation and investigation of the smooth Voronoi potential (2) implementation of CAP in the projected scheme, and (3) development of the OpenCAP package, which extends the CAP methodology to popular electronic structure packages. The second major focus is the identification of electron and hole transfer (ET) pathways in biomolecules. Both experimental and theoretical inquiries into electron/hole transfer processes in biomolecules generally require targeted approaches, which are complicated by the existence of numerous potential pathways. To this end, we have developed an open-source web platform, eMap, which exploits a coarse-grained model of the protein crystal structure to (1) enable pre-screening of potentially efficient ET pathways, and (2) identify shared pathways/motifs in families of proteins. Following introductory chapters on motivation and theoretical background, we devote a chapter to each new methodology mentioned above. The open-source software tools discussed herein are under active development, and have been utilized in published work by several unaffiliated experimental and theoretical groups across the world. We conclude the dissertation with a summary and discussion of the outlook and future directions of the OpenCAP and eMap software packages.

Lozenges are polygons constructed by gluing two equilateral triangles along an edge. We can fit lozenge pieces together to form larger polygons and given an appropriate polygon we can tile it with lozenges. Lozenge tilings of the semi-regular hexagon with sides A,B,C can be viewed as the 2D picture of a stack of cubes in a A x B x C box. In this project we investigate the typical shape of a tiling as the sides A,B,C of the box grow uniformly to infinity and we consider two cases: The uniform case where all tilings occur with equal probability and the q^Volume case where the probability of a tiling is proportional to the volume taken up by the corresponding stack of cubes. To investigate lozenge tilings we transform it into a question on families of non-intersecting paths on a corresponding graph representing the hexagon. Using the Lindström–Gessel–Viennot theorem we can define the probability of a non-intersecting path crossing a particular point in the hexagon both for the uniform and the $q$-Volume case. In each case this probability function is connected to either the Hahn or the $q$-Hahn orthogonal polynomials. The orthogonal polynomials depend on the sides of the hexagon and so we consider the asymptotic behaviour of the polynomials as the sides grow to infinity using a result due to Kuijlaars and Van Assche. This determines the density of non-intersecting paths through every point in the hexagon, which we calculate, and a ``Arctic curve" result which shows that the six corners of the hexagon are (with probability one) tiled with just one type of lozenge. / "Lozenger" är polygoner konstruerade genom att limma två liksidiga trianglar längs en kant. Vi kan montera lozengstycken ihop för att bilda större polygoner och med en lämplig polygon kan vi lozengplatta den. Lozengplattor av den semi-liksidiga hexagonen med sidorna A, B, C kan ses som 2D-bilden av en stapel kuber i en A x B x C-box. I det här projektet undersöker vi den typiska formen på en platta när sidorna A, B, C på rutan växer till oändlighet och vi tar an två fall: Det likformiga fallet där alla plattor sker med samma sannolikhet och q ^ Volymfallet då sannolikheten för en platta är proportionell mot volymen som tas upp av motsvarande kubstapel. För att undersöka plattor förvandlar vi det till en fråga om samlingar av icke-korsande vägar på en motsvarande graf som representerar hexagonen. Med hjälp av satsen Lindström – Gessel – Viennot kan vi definiera sannolikheten för att en icke-korsande väg går genom en viss punkt i hexagonen både för det enhetliga och $ q $ -volymfallet. I båda fallen är dessa sannolikhetsfunktioner relaterade till Hahn eller $ q $ -Hahn ortogonala polynomer. Dessa ortogonala polynom beror på hexagonens sidor så vi betraktar polynomens asymptotiska beteende när sidorna växer till oändlighet genom ett resultat från Kuijlaars och Van Assche. Detta bestämmer densiteten för de icke-korsande vägarna genom varje punkt i det hexagon vi beräknar. Detta bestämmer också också en '' arktisk kurva '' som visar att hexagonens sex hörn är (med sannolikhet ett) plattade med bara en typ av lozeng.

The rational design and construction of Metal-Organic Frameworks (MOFs) with intricate structural complexity are of prime importance in reticular chemistry. However, the design of intricate structures that can practically be synthesized is very difficult, and the suitable targeted intricate nets are still unexplored. Evidently, it is of great value to build the fundamental theory for the design of intricate structures. This dissertation is focused on the exploration of cutting-edge design methodologies in reticular chemistry. This research shows the design and synthesis of several MOF platforms (hex, fcu, gea and the) based on rare earth polynuclear clusters. Furthermore, this research unveils the latest addition, named merged nets approach, to the design toolbox in reticular chemistry for the rational design and construction of intricate mixed-linker MOFs. In essence, a valuable net for design enclosing two edges is rationally generated by merging two edge-transitive nets, spn and hxg. The resultant merged net, named sph net, offers potential for the deliberate design and construction of highly symmetric isoreticular intricate mixed-linker MOFs, sph-MOF-1 to 4, which represent the first examples of MOFs where the underlying net is merged from two 3-periodic edge-transitive nets. Furthermore, the underlying principle of the merged net approach, the fundamental merged net equation, and two key parameters are disclosed. Also, we discovered three analysis methods to check and validate corresponding signature nets in an edge-transitive net. Based on these analysis methods, a signature map of all edge-transitive nets was established. This map showing the systematic relationship among edge-transitive nets will help the material chemist to comprehend more about the underlying nets in reticular chemistry. Based on the revealed map, we systematically described the nine types of merging combination and 140 merged nets based on two edge-transitive nets. Among these enumerated nets, only 18 of them was shown on the RCSR database before. These enumerated merged nets significantly increased the designable targets in reticular chemistry. Using an example of enumerated sub net, we show how this approach can be utilized to design and synthesis mixed-linker porous materials based on the intricate sub-MOF platform, which presents one of the most intricate MOF structures synthesized by design.