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In this paper, we provide a unified definition of mediated graph, a combinatorial structure with multiple applications in mathematical optimization. We study some geometric and algebraic properties of this family of graphs and analyze…
Time-varying graph signals are alternative representation of multivariate (or multichannel) signals in which a single time-series is associated with each of the nodes or vertex of a graph. Aided by the graph-theoretic tools, time-varying…
Graph signal processing has become an essential tool for analyzing data structured on irregular domains. While conventional graph shift operators (GSOs) are effective for certain tasks, they inherently lack flexibility in modeling…
Graphons are infinite-dimensional objects that represent the limit of convergent sequences of graphs as their number of nodes goes to infinity. This paper derives a theory of graphon signal processing centered on the notions of graphon…
A non-negative integer invariant, estimating from below the number of geometrically different critical points of a smooth function $f$ defined in the 2-disk, $f:\mathbb{B}^{2}\rightarrow\mathbb{R}$, is considered. (We denote it by…
We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…
Let $\Gamma$ be a finite, undirected, connected, simple graph. We say that a matching $\mathcal{M}$ is a \textit{permutable $m$-matching} if $\mathcal{M}$ contains $m$ edges and the subgroup of $\text{Aut}(\Gamma)$ that fixes the matching…
Graph transformers are a recent advancement in machine learning, offering a new class of neural network models for graph-structured data. The synergy between transformers and graph learning demonstrates strong performance and versatility…
Quantifying the complexity of large graphs requires measures that extend beyond predefined structural features and scale efficiently with graph size. This work adopts a generative perspective, modeling large networks as exchangeable graphs…
We present a unified framework to describe lattice gauge theories by means of tensor networks: this framework is efficient as it exploits the high amount of local symmetry content native of these systems describing only the gauge invariant…
Local gauge structures play a central role in a wide range of condensed matter systems and synthetic quantum platforms, where they emerge as effective descriptions of strongly correlated phases and engineered dynamics. We introduce a…
We study invariants of virtual graphoids, which are virtual spatial graph diagrams with two distinguished degree-one vertices modulo graph Reidemeister moves applied away from the distinguished vertices. Generalizing previously known…
We classify the invariant Borel measures for adic transformations, where the alphabets have bounded size and the measure is finite on the path space of some sub-Bratteli diagram. We develop a nonstationary version of the Frobenius normal…
Graph transformation formalisms have proven to be suitable tools for the modelling of chemical reactions. They are well established in theoretical studies and increasingly also in practical applications in chemistry. The latter is made…
In this paper, we introduce a graph matching method that can account for constraints of arbitrary order, with arbitrary potential functions. Unlike previous decomposition approaches that rely on the graph structures, we introduce a…
Recently a permutation on Dyck paths, related to the chip firing game, was introduced and studied by Barnabei et al.. It is called $\gamma$-operator, and uses symmetries and reflections to relate Dyck paths having the same length. A…
We present new combinatorial objects, which we call grid-labelled graphs, and show how these can be used to represent the quantum states arising in a scenario which we refer to as the faulty emitter scenario: we have a machine designed to…
Generative models of graph structure have applications in biology and social sciences. The state of the art is GraphRNN, which decomposes the graph generation process into a series of sequential steps. While effective for modest sizes, it…
An $L(2,1)$-labelling of a finite graph $\Gamma$ is a function that assigns integer values to the vertices $V(\Gamma)$ of $\Gamma$ (colouring of $V(\Gamma)$ by ${\mathbb{Z}}$) so that the absolute difference of two such values is at least…
A surface in a three-dimensional metric Lie group $G$ is said invariant if it is invariant with respect to a one-dimensional subgroup $\Gamma$ of the isometry group of $G$. Is this work we focus on unimodular metric Lie groups $G$ that can…