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Analyzing nodal domains is a way to discern the structure of eigenvectors of operators on a graph. We give a new definition extending the concept of nodal domains to arbitrary signed graphs, and therefore to arbitrary symmetric matrices. We…
A connected undirected graph $G = (V,E)$ is lower conformally rigid if uniform edge weights maximize the second smallest Laplacian eigenvalue $\lambda_2(w)$ over all normalized edge weights $w$, and upper conformally rigid if uniform edge…
Many natural and social systems develop complex networks, that are usually modelled as random graphs. The eigenvalue spectrum of these graphs provides information about their structural properties. While the semi-circle law is known to…
The corona of hypergraphs is an extension of the corona operation applied to graphs. The corona $G_0^* \odot_1^n G_1^*$ of two hypergraphs is obtained by taking $n$ copies of $G_1^*$ (where $n$ is the order of $G_0^*$) and by joining the…
Graph Neural Networks (GNNs) have received increasing attention in many fields. However, due to the lack of prior graphs, their use for semantic labeling has been limited. Here, we propose a novel architecture called the Self-Constructing…
The spectral theory of graphs provides a bridge between classical signal processing and the nascent field of graph signal processing. In this paper, a spectral graph analogy to Heisenberg's celebrated uncertainty principle is developed.…
Graph convolutional networks (GCNs) aim at extending deep learning to arbitrary irregular domains, namely graphs. Their success is highly dependent on how the topology of input graphs is defined and most of the existing GCN architectures…
Spectral clustering is a well-known technique which identifies $k$ clusters in an undirected graph with weight matrix $W\in\mathbb{R}^{n\times n}$ by exploiting its graph Laplacian $L(W)$, whose eigenvalues $0=\lambda_1\leq \lambda_2 \leq…
We formulate confinement in QCD as an entropic surface phenomenon. Quark and gluon quantum information is localized on a transverse entangling two-sphere of radius $R_{EE}$; at this radius the QCD vacuum -- partitioned by a hadron into…
A metrized graph is a compact singular 1-manifold endowed with a metric. A given metrized graph can be modelled by a family of weighted combinatorial graphs. If one chooses a sequence of models from this family such that the vertices become…
Laplacian eigenvectors capture natural community structures on graphs and are widely used in spectral clustering and manifold learning. The use of Laplacian eigenvectors as embeddings for the purpose of multiscale graph comparison has…
Network scientists have shown that there is great value in studying pairwise interactions between components in a system. From a linear algebra point of view, this involves defining and evaluating functions of the associated adjacency…
We introduce a class of random graphs with a community structure, which we call the hierarchical configuration model. On the inter-community level, the graph is a configuration model, and on the intra-community level, every vertex in the…
We consider the problem of estimating graph limits, known as graphons, from observations of sequences of sparse finite graphs. In this paper we show a simple method that can shed light on a subset of sparse graphs. The method involves…
We present a mathematical framework for describing the topology of configuration spaces for particles on one-connected graphs. In particular, we compute the homology groups over integers for different classes of one-connected graphs. Our…
We establish spectral rigidity for spherically symmetric manifolds with boundary and interior interfaces determined by discontinuities in the metric under certain conditions. Rather than a single metric, we allow two distinct metrics in…
Graph embedding techniques are useful to characterize spectral signature relations for hyperspectral images. However, such images consists of disjoint classes due to spatial details that are often ignored by existing graph computing tools.…
In hyperspectral image (HSI) classification, spatial context has demonstrated its significance in achieving promising performance. However, conventional spatial context-based methods simply assume that spatially neighboring pixels should…
In this study, we explore the interrelation between hypergraph symmetries represented by equivalence relations on the vertex set and the spectra of operators associated with the hypergraph. We introduce the idea of equivalence relation…
Semantic segmentation has innately relied on extensive pixel-level annotated data, leading to the emergence of unsupervised methodologies. Among them, leveraging self-supervised Vision Transformers for unsupervised semantic segmentation…