Related papers: Re-imagining Spectral Graph Theory
A signed graph product is defined for a new product, and initially the unsigned graph product's Laplacian spectrum and signless Laplacian spectrum are found. Next, for the signed graph product, the adjacency spectrum, Laplacian spectrum,…
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…
In this note we elaborate on some notions of surface area for discrete graphs which are closely related to the inverse degree. These notions then naturally lead to associated connectivity measures of graphs and to the definition of a…
The data clustering problem consists in dividing a data set into prescribed groups of homogeneous data. This is a NP-hard problem that can be relaxed in the spectral graph theory, where the optimal cuts of a graph are related to the…
Consider two graphs $G$ and $H$. Let $H^k[G]$ be the lexicographic product of $H^k$ and $G$, where $H^k$ is the lexicographic product of the graph $H$ by itself $k$ times. In this paper, we determine the spectrum of $H^k[G]$ and $H^k$ when…
In this paper, we determine the two normalized Laplacian spectrum of generalized subdivision- vertex corona, subdivision-edge corona for a connected regular graph with an arbitrary reg- ular graph in terms of their normalized Laplacian…
Graph neural networks (GNNs) are fundamental tools in graph machine learning. The performance of GNNs relies crucially on the availability of informative node features, which can be limited or absent in real-life datasets and applications.…
It is conjectured that the Laplacian spectrum of a graph is majorized by its conjugate degree sequence. In this paper, we prove that this majorization holds for a class of graphs including trees. We also show that a generalization of this…
Learning the graph Laplacian from observed data is one of the most investigated and fundamental tasks in Graph Signal Processing (GSP). Different variants of the Laplacian, such as the combinatorial, signless or signed Laplacians have been…
In data science, hypergraphs are natural models for data exhibiting multi-way relations, whereas graphs only capture pairwise. Nonetheless, many proposed hypergraph neural networks effectively reduce hypergraphs to undirected graphs via…
We consider Laplacians on $\Z^2$-periodic discrete graphs. The following results are obtained: 1) The Floquet-Bloch decomposition is constructed and basic properties are derived. 2) The estimates of the Lebesgue measure of the spectrum in…
This paper surveys some recent results and progress on the extremal prob- lems in a given set consisting of all simple connected graphs with the same graphic degree sequence. In particular, we study and characterize the extremal graphs…
In this article we examine the adjacency and Laplacian matrices and their eigenvalues and energies of the general product (non-complete extended $p$-sum, or NEPS) of signed graphs. We express the adjacency matrix of the product in terms of…
Let G be a graph on n vertices. The Laplacian matrix of G, denoted by L(G), is defined as L(G) = D(G) - A(G), where A(G) is the adjacency matrix of G and D(G) is the diagonal matrix of the vertex degrees of G. A graph G is said to be…
Spectral approaches of network analysis heavily rely upon the eigendecomposition of the graph Laplacian. For instance, in graph signal processing, the Laplacian eigendecomposition is used to define the graph Fourier transform and then…
This paper investigates the use of methods from partial differential equations and the Calculus of variations to study learning problems that are regularized using graph Laplacians. Graph Laplacians are a powerful, flexible method for…
Kusuoka and Zhou have defined the Laplacian on the Sierpinski carpet using average values of functions on small cells and the graph structure of cell adjacency. We have implemented an algorithm that uses their method to approximate…
The kernel of the natural projection of a graph product of groups onto their direct product is called the Cartesian subgroup of the graph product. This construction generalises commutator subgroups of right-angled Coxeter and Artin groups.…
The aim of the present paper is to analyse the spectrum of Laplace and Dirac type operators on metric graphs. In particular, we show for equilateral graphs how the spectrum (up to exceptional eigenvalues) can be described by a natural…
For a given graph $G$, we aim to determine the possible realizable spectra for a generalized (or sometimes referred to as a weighted) Laplacian matrix associated with $G$. This new specialized inverse eigenvalue problem is considered for…