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This paper investigates the enumeration of Cayley digraphs, focusing on counting Cayley digraphs on dihedral groups up to CI-isomorphism. By leveraging the Cauchy-Frobenius Lemma and properties of automorphisms, we derive an explicit…
Precise knowledge of causal directed acyclic graphs (DAGs) is assumed for standard approaches towards valid adjustment set selection for unbiased estimation, but in practice, the DAG is often inferred from data or expert knowledge,…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
Deep generative models have achieved great success in areas such as image, speech, and natural language processing in the past few years. Thanks to the advances in graph-based deep learning, and in particular graph representation learning,…
Causal structure learning from observational data remains a non-trivial task due to various factors such as finite sampling, unobserved confounding factors, and measurement errors. Constraint-based and score-based methods tend to suffer…
In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$.
In the past few decades, quantum algorithms have become a popular research area of both mathematicians and engineers. Among them, uniform mixing provides a uniform probability distribution of quantum information over time which attracts a…
A directed hypergraph (dihypergraph) consists of a set of vertices and a set of hyperarcs, where each hyperarc is partitioned into a head and a tail. Directed hypergraphs are useful in many applications, including the study of chemical…
Exploring small connected and induced subgraph patterns (CIS patterns, or graphlets) has recently attracted considerable attention. Despite recent efforts on computing the number of instances a specific graphlet appears in a large graph…
Directed mixed graphs permit directed and bidirected edges between any two vertices. They were first considered in the path analysis developed by Sewall Wright and play an essential role in statistical modeling. We introduce a matrix…
Recent work on the structure of social networks and the internet has focussed attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in…
We construct a strongly regular graph with the parameters (65; 32; 15; 16). The idea is to search for an adjacency matrix that consists of circulant blocks. Equations with such matrices can be reduced to congruences with polynomials…
Let $p$ be an odd prime, and $D_{2p}=\langle \tau,\sigma\mid \tau^p=\sigma^2=e,\sigma\tau\sigma=\tau^{-1}\rangle$ the dihedral group of order $2p$. In this paper, we provide the number of (connected) Cayley (di-)graphs on $D_{2p}$ up to…
We consider the normalized Laplace operator for directed graphs with positive and negative edge weights. This generalization of the normalized Laplace operator for undirected graphs is used to characterize directed acyclic graphs. Moreover,…
The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of…
The structure of many real networks is not locally tree-like and hence, network analysis fails to characterise their bond percolation properties. In a recent paper [P. Mann, V. A. Smith, J. B. O. Mitchell, and S. Dobson, Percolation in…
Many scientific datasets are of high dimension, and the analysis usually requires visual manipulation by retaining the most important structures of data. Principal curve is a widely used approach for this purpose. However, many existing…
In this contribution we present a construction of large networks of diameter two and of order $\frac{1}{2}d^2$ for every degree $d\geq 8$, based on Cayley graphs with surprisingly simple underlying groups. For several small degrees we…
We investigate the structure of conformally rigid graphs. Graphs are conformally rigid if introducing edge weights cannot increase (decrease) the second (last) eigenvalue of the Graph Laplacian. Edge-transitive graphs and distance-regular…
Traditional graph analysis focuses on nodes and edges, that is, pairwise relationships. Yet many real-world networks, including biological, social, and communication networks, involve higher-order relationships in which multiple nodes…