PP-polynomial weakly distance-regular digraphs

A weakly distance-regular digraph is $P$-polynomial if its attached scheme is $P$-polynomial. In this paper, we characterize all $P$-polynomial weakly distance-regular digraphs

Weakly distance-regular circulants, I

We classify certain non-symmetric commutative association schemes. As an application, we determine all the weakly distance-regular circulants of one type of arcs by using Schur rings. We also give the classification of primitive weakly distance-regular circulants.Comment: 28 page

Bipartite Quantum Walks and the Hamiltonian

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

Extremal Directed And Mixed Graphs

We consider three problems in extremal graph theory, namely the degree/diameter problem, the degree/geodecity problem and Tur\'{a}n problems, in the context of directed and partially directed graphs. A directed graph or mixed graph $G$ is $k$-geodetic if there is no pair of vertices $u,v$ of $G$ such that there exist distinct non-backtracking walks with length $\leq k$ in $G$ from $u$ to $v$. The order of a $k$-geodetic digraph with minimum out-degree $d$ is bounded below by the \emph{directed Moore bound} $M(d,k) = 1+d+d^2+\dots +d^k$; similarly the order of a $k$-geodetic mixed graph with minimum undirected degree $r$ and minimum directed out-degree $z$ is bounded below by the \emph{mixed Moore bound}. We will be interested in networks with order exceeding the Moore bound by some small \emph{excess} $\epsilon$. The \emph{degree/geodecity problem} asks for the smallest possible order of a $k$-geodetic digraph or mixed graph with given degree parameters. We prove the existence of extremal graphs, which we call \emph{geodetic cages}, and provide some bounds on their order and information on their structure. We discuss the structure of digraphs with excess one and rule out the existence of certain digraphs with excess one. We then classify all digraphs with out-degree two and excess two, as well as all diregular digraphs with out-degree two and excess three. We also present the first known non-trivial examples of geodetic cages. We then generalise this work to the setting of mixed graphs. First we address the question of the total regularity of mixed graphs with order close to the Moore bound and prove bounds on the order of mixed graphs that are not totally regular. In particular using spectral methods we prove a conjecture of L\'{o}pez and Miret that mixed graphs with diameter two and order one less than the Moore bound are totally regular. Using counting arguments we then provide strong bounds on the order of totally regular $k$-geodetic mixed graphs and use these results to derive new extremal mixed graphs. Finally we change our focus and study the Tur\'{a}n problem of the largest possible size of a $k$-geodetic digraph with given order. We solve this problem and also prove an exact expression for the restricted problem of the largest possible size of strongly connected $2$-geodetic digraphs, as well as providing constructions of strongly connected $k$-geodetic digraphs that we conjecture to be extremal for larger $k$. We close with a discussion of some related generalised Tur\'{a}n problems for $k$-geodetic digraphs

Non-topological persistence for data analysis and machine learning

This thesis main objective is to study possible applications of the generalisation of persistence theory introduced in [1], [2]. This generalisation extends the notion of persistence to a wider categorical setting, avoiding constructing secondary structures as topological spaces. The first field analysed is graph theory. At first, we studied which classical graph theory invariants could be used as rank function. Another aspect analysed in this thesis is the extension of the study of connectivity in graphs from a persistence viewpoint started in [1] to oriented graphs. Moreover, we studied how different orientation of the same underlying graph can change the distribution of cornerpoints in persistence diagrams, both in deterministic and random graphs. The other application field analysed is image processing. We adapted the notion of steady and ranging sets to the category of sets and used them to define activation and deactivation rules for each pixel. These notions allowed us to define a filter capable of enhancing the signal of pixels close to a border. This filter has proven to be stable under salt and pepper noise perturbation. At last, we used this filter to define a novel pooling layer for convolutional neural networks. In the experimental part, we compared the proposed layer with other state-of-the-art layers. The results show how the proposed layer outperform the other layers in term of accuracy. Moreover, by concatenating the proposed and the Max pooling, it is possible to improve accuracy further. [1] Bergomi, M.G., Ferri, M., Vertechi, P., Zuffi, L. (2020), Beyond topological persistence: Starting from networks, arXiv. [2] Bergomi, M. G., & Vertechi, P. (2020). Rank-based persistence. Theory and Applications of Categories, 35, 228-260

Combinatorics

Combinatorics is a fundamental mathematical discipline that focuses on the study of discrete objects and their properties. The present workshop featured research in such diverse areas as Extremal, Probabilistic and Algebraic Combinatorics, Graph Theory, Discrete Geometry, Combinatorial Optimization, Theory of Computation and Statistical Mechanics. It provided current accounts of exciting developments and challenges in these fields and a stimulating venue for a variety of fruitful interactions. This is a report on the meeting, containing extended abstracts of the presentations and a summary of the problem session