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A foundation model like GPT elicits many emergent abilities, owing to the pre-training with broad inclusion of data and the use of the powerful Transformer architecture. While foundation models in natural languages are prevalent, can we…
We generalize the class of split graphs to the directed case and show that these split digraphs can be identified from their degree sequences. The first degree sequence characterization is an extension of the concept of splittance to…
We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction…
Some of the basic properties of any dynamical system can be summarized by a graph. The dynamical systems in our theory run from maps like the logistic map to ordinary differential equations to dissipative partial differential equations. Our…
Model repair is an essential topic in model-driven engineering. Since models are suitably formalized as graph-like structures, we consider the problem of rule-based graph repair: Given a rule set and a graph constraint, try to construct a…
Knowing when a graphical model is perfect to a distribution is essential in order to relate separation in the graph to conditional independence in the distribution, and this is particularly important when performing inference from data.…
A typical result in graph theory says that a graph $G$, satisfying certain conditions, has some property $\cal P$. Once such a theorem is established, it is natural to ask how strongly $G$ satisfies $\cal P$. Can one strengthen the result…
Graph Convolutional Networks (GCNs) have been widely used due to their outstanding performance in processing graph-structured data. However, the undirected graphs limit their application scope. In this paper, we extend spectral-based graph…
Sampling random graphs is essential in many applications, and often algorithms use Markov chain Monte Carlo methods to sample uniformly from the space of graphs. However, often there is a need to sample graphs with some property that we are…
In the past two decades, significant advances have been made in understanding the structural and functional properties of biological networks, via graph-theoretic analysis. In general, most graph-theoretic studies are conducted in the…
We present, to the best of the authors' knowledge, all known results for the (planar) crossing numbers of specific graphs and graph families. The results are separated into various categories; specifically, results for general graph…
The dynamics of coupled intermittent maps is used to model the correlated structure of genomic sequences. The use of intermittent maps, as opposed to other simple chaotic maps, is particularly suited for the production of long range…
Vector fields with components which are generalized zero-forms are constructed. Inner products with generalized forms, Lie derivatives and Lie brackets are computed. The results are shown to generalize previously reported results for…
We characterize the computational power of neural networks that follow the graph neural network (GNN) architecture, not restricted to aggregate-combine GNNs or other particular types. We establish an exact correspondence between the…
It was recently proved that every planar graph is a subgraph of the strong product of a path and a graph with bounded treewidth. This paper surveys generalisations of this result for graphs on surfaces, minor-closed classes, various…
Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that,…
Compositional generalization is a crucial property in artificial intelligence, enabling models to handle novel combinations of known components. While most deep learning models lack this capability, certain models succeed in specific tasks,…
Owing to their versatility, graph structures admit representations of intricate relationships between the separate entities comprising the data. We formalise the notion of connection between two vertex sets in terms of edge and vertex…
Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in…
We have defined and established a theory of cofinite connectedness of a cofinite graph. Many of the properties of connectedness of topological spaces have analogs for cofinite connectedness. We have seen that if $G$ is a cofinite group and…