Related papers: Equivalent spectral theory for fundamental graph c…
The aim of this paper is twofold: (1) On the one hand, the paper revisits the spectral analysis of semigroups in a general Banach space setting. It presents some new and more general versions, and provides comprehensible proofs, of…
Algorithms based on spectral graph cut objectives such as normalized cuts, ratio cuts and ratio association have become popular in recent years because they are widely applicable and simple to implement via standard eigenvector…
We study the problem of efficiently refuting the k-colorability of a graph, or equivalently certifying a lower bound on its chromatic number. We give formal evidence of average-case computational hardness for this problem in sparse random…
Many common methods for data analysis rely on linear algebra. We provide new results connecting data analysis error to numerical accuracy, which leads to the first meaningful stopping criterion for two way spectral partitioning. More…
We show that the joint spectral radius of a finite collection of nonnegative matrices can be bounded by the eigenvalue of a non-linear operator. This eigenvalue coincides with the ergodic constant of a risk-sensitive control problem, or of…
We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph…
We define a range of new coarse geometric invariants based on various graph-theoretic measures of complexity for finite graphs, including: treewidth, pathwidth, cutwidth and bandwidth. We prove that, for bounded degree graphs, these…
Image segmentation is a fundamental research topic in image processing and computer vision. In the last decades, researchers developed a large number of segmentation algorithms for various applications. Amongst these algorithms, the…
Interest in inverse dynamical, spectral and scattering problems for differential equations on graphs is motivated by possible applications to nano-electronics and quantum waveguides and by a variety of other classical and quantum…
Given a simple graph $G$, its Laplacian-energy-like invariant $LEL(G)$ and incidence energy $IE(G)$ are the sum of square root of its all Laplacian eigenvalues and signless Laplacian eigenvalues, respectively. Applying the Cauchy-Schwarz…
Contrastive learning has emerged as a premier method for learning representations with or without supervision. Recent studies have shown its utility in graph representation learning for pre-training. Despite successes, the understanding of…
We study the problem of determining the optimal low dimensional projection for maximising the separability of a binary partition of an unlabelled dataset, as measured by spectral graph theory. This is achieved by finding projections which…
Normalized-cut graph partitioning aims to divide the set of nodes in a graph into $k$ disjoint clusters to minimize the fraction of the total edges between any cluster and all other clusters. In this paper, we consider a fair variant of the…
The fundamental gap is the difference between the first two Dirichlet eigenvalues of a Schr\"odinger operator (and the Laplacian, in particular). For horoconvex domains in hyperbolic space, Nguyen, Stancu and Wei conjectured that it is…
Partitioning the vertices of a graph into two roughly equal parts while minimizing the number of edges crossing the cut is a fundamental problem (called Balanced Separator) that arises in many settings. For this problem, and variants such…
We present a rectangle-based segmentation algorithm that sets up a graph and performs a graph cut to separate an object from the background. However, graph-based algorithms distribute the graph's nodes uniformly and equidistantly on the…
The eigenvectors for graph $1$-Laplacian possess some sort of localization property: On one hand, any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a…
Graph similarity computation is one of the core operations in many graph-based applications, such as graph similarity search, graph database analysis, graph clustering, etc. Since computing the exact distance/similarity between two graphs…
We prove that the graph C*-algebra $C^*(E)$ of a trimmable graph $E$ is $U(1)$-equivariantly isomorphic to a pullback C*-algebra of a subgraph C*-algebra $C^*(E'')$ and the C*-algebra of functions on a circle tensored with another subgraph…
The Cheeger constant of a graph, or equivalently its coboundary expansion, quantifies the expansion of the graph. This notion assumes an implicit choice of a coefficient group, namely, $\mathbb{F}_2$. In this paper, we study Cheeger-type…