Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

Graph Coverings with Few Eigenvalues or No Short Cycles

This thesis addresses the extent of the covering graph construction. How much must a cover X resemble the graph Y that it covers? How much can X deviate from Y? The main statistics of X and Y which we will measure are their regularity, the spectra of their adjacency matrices, and the length of their shortest cycles. These statistics are highly interdependent and the main contribution of this thesis is to advance our understanding of this interdependence. We will see theorems that characterize the regularity of certain covering graphs in terms of the number of distinct eigenvalues of their adjacency matrices. We will see old examples of covers whose lack of short cycles is equivalent to the concentration of their spectra on few points, and new examples that indicate certain limits to this equivalence in a more general setting. We will see connections to many combinatorial objects such as regular maps, symmetric and divisible designs, equiangular lines, distance-regular graphs, perfect codes, and more. Our main tools will come from algebraic graph theory and representation theory. Additional motivation will come from topological graph theory, finite geometry, and algebraic topology

Cometric Association Schemes

The combinatorial objects known as association schemes arise in group theory, extremal graph theory, coding theory, the design of experiments, and even quantum information theory. One may think of a d-class association scheme as a (d + 1)-dimensional matrix algebra over R closed under entrywise products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed under the entrywise product. Such systems of imprimitivity provide us with quotient schemes, smaller association schemes which are often easier to understand, providing useful information about the structure of the larger scheme. One important property of any association scheme is that we may find a basis of d + 1 idempotent matrices for our algebra. A cometric scheme is one whose idempotent basis may be ordered E0, E1, . . . , Ed so that there exists polynomials f0, f1, . . . , fd with fi ◦ (E1) = Ei and deg(fi) = i for each i. Imprimitive cometric schemes relate closely to t-distance sets, sets of unit vectors with only t distinct angles, such as equiangular lines and mutually unbiased bases. Throughout this thesis we are primarily interested in three distinct goals: building new examples of cometric association schemes, drawing connections between cometric association schemes and other objects either combinatorial or geometric, and finding new realizability conditions on feasible parameter sets — using these conditions to rule out open parameter sets when possible. After introducing association schemes with relevant terminology and definitions, this thesis focuses on a few recent results regarding cometric schemes with small d. We begin by examining the matrix algebra of any such scheme, first looking for low rank positive semidefinite matrices with few distinct entries and later establishing new conditions on realizable parameter sets. We then focus on certain imprimitive examples of both 3- and 4-class cometric association schemes, generating new examples of the former while building realizability conditions for both. In each case, we examine the related t-distance sets, giving conditions which work towards equivalence; in the case of 3-class Q-antipodal schemes, an equivalence is established. We conclude by partially extending a result of Brouwer and Koolen concerning the connectivity of graphs arising from metric association schemes

Equiangular Lines and Antipodal Covers

It is not hard to see that the number of equiangular lines in a complex space of dimension $d$ is at most $d^{2}$. A set of $d^{2}$ equiangular lines in a $d$-dimensional complex space is of significant importance in Quantum Computing as it corresponds to a measurement for which its statistics determine completely the quantum state on which the measurement is carried out. The existence of $d^{2}$ equiangular lines in a $d$-dimensional complex space is only known for a few values of $d$, although physicists conjecture that they do exist for any value of $d$. The main results in this thesis are: \begin{enumerate} \item Abelian covers of complete graphs that have certain parameters can be used to construct sets of $d^2$ equiangular lines in $d$-dimen\-sion\-al space; \item we exhibit infinitely many parameter sets that satisfy all the known necessary conditions for the existence of such a cover; and \item we find the decompose of the space into irreducible modules over the Terwilliger algebra of covers of complete graphs. \end{enumerate} A few techniques are known for constructing covers of complete graphs, none of which can be used to construct covers that lead to sets of $d^{2}$ equiangular lines in $d$-dimensional complex spaces. The third main result is developed in the hope of assisting such construction

Discrete curvatures motivated from Riemannian geometry and optimal transport: Bonnet-Myers-type diameter bounds and rigidity

This thesis gives an overview of three notions of Ricci curvature for discrete spaces, including Ollivier Ricci curvature (motivated from optimal transport), Bakry-{\'E}mery curvature (from Bochner’s formula in Riemannian geometry) and Erbar-Maas entropic Ricci curvature (from optimal transport). The first part of the thesis provides background knowledge in optimal transport theory and Riemannian geometry which is essential to the understanding of generalized Ricci curvatures for metric measure spaces and the mentioned Ricci curvatures for graphs. For each of the three discrete curvature notions, discussed in their respective part of the thesis, we provide the definition of the curvature and use hypercubes as an example for the curvature calculation. We study various curvature results with an emphasis on upper bounds of diameter and lower bounds of the spectral gap for graphs with positive lower bound on the Ricci curvature. These results can be regarded as discrete analogues of the Bonnet-Myers theorem and the Lichnerowicz theorem in Riemannian geometry. In addition, we deeply investigate into the rigidity results (analogous to Cheng’s rigidity) in attempt to classify all graphs which yield the sharp diameter bound in the sense of Ollivier Ricci curvature and Bakry-{\'E}mery curvature

LIPIcs, Volume 251, ITCS 2023, Complete Volume

LIPIcs, Volume 251, ITCS 2023, Complete Volum

Foundations of Software Science and Computation Structures

This open access book constitutes the proceedings of the 23rd International Conference on Foundations of Software Science and Computational Structures, FOSSACS 2020, which took place in Dublin, Ireland, in April 2020, and was held as Part of the European Joint Conferences on Theory and Practice of Software, ETAPS 2020. The 31 regular papers presented in this volume were carefully reviewed and selected from 98 submissions. The papers cover topics such as categorical models and logics; language theory, automata, and games; modal, spatial, and temporal logics; type theory and proof theory; concurrency theory and process calculi; rewriting theory; semantics of programming languages; program analysis, correctness, transformation, and verification; logics of programming; software specification and refinement; models of concurrent, reactive, stochastic, distributed, hybrid, and mobile systems; emerging models of computation; logical aspects of computational complexity; models of software security; and logical foundations of data bases.

Defining complex rule-based models in space and over time

Computational biology seeks to understand complex spatio-temporal phenomena across multiple levels of structural and functional organisation. However, questions raised in this context are difficult to answer without modelling methodologies that are intuitive and approachable for non-expert users. Stochastic rule-based modelling languages such as Kappa have been the focus of recent attention in developing complex biological models that are nevertheless concise, comprehensible, and easily extensible. We look at further developing Kappa, in terms of how we might define complex models in both the spatial and the temporal axes. In defining complex models in space, we address the assumption that the reaction mixture of a Kappa model is homogeneous and well-mixed. We propose evolutions of the current iteration of Spatial Kappa to streamline the process of defining spatial structures for different modelling purposes. We also verify the existing implementation against established results in diffusion and narrow escape, thus laying the foundations for querying a wider range of spatial systems with greater confidence in the accuracy of the results. In defining complex models over time, we draw attention to how non-modelling specialists might define, verify, and analyse rules throughout a rigorous model development process. We propose structured visual methodologies for developing and maintaining knowledge base data structures, incorporating the information needed to construct a Kappa rule-based model. We further extend these methodologies to deal with biological systems defined by the activity of synthetic genetic parts, with the hope of providing tractable operations that allow multiple users to contribute to their development over time according to their area of expertise. Throughout the thesis we pursue the aim of bridging the divide between information sources such as literature and bioinformatics databases and the abstracting decisions inherent in a model. We consider methodologies for automating the construction of spatial models, providing traceable links from source to model element, and updating a model via an iterative and collaborative development process. By providing frameworks for modellers from multiple domains of expertise to work with the language, we reduce the entry barrier and open the field to further questions and new research