Related papers: An isoperimetric constant for signed graphs
Several new spectral properties of the normalized Laplacian defined for oriented hypergraphs are shown. The eigenvalue $1$ and the case of duplicate vertices are discussed; two Courant nodal domain theorems are established; new quantities…
We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1-Laplacian ?$\Delta_1$. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the…
In this paper we define signless Laplacian matrix of a hypergraph and obtain structural properties from its eigenvalues. We generalize several known results for graphs, relating the spectrum of this matrix with structural parameters of the…
We present a systematic collection of spectral surgery principles for the Laplacian on a metric graph with any of the usual vertex conditions (natural, Dirichlet or $\delta$-type), which show how various types of changes of a local or…
We propose the notion of normalized Laplacian matrix $\mathcal{L}(\Phi)$ for a gain graphs and study its properties in detail, providing insights and counterexamples along the way. We establish bounds for the eigenvalues of…
We prove new properties of the non-backtracking graph and the non-backtracking Laplacian for graphs. In particular, among other results, we prove that two simple graphs are isomorphic if and only if their corresponding non-backtracking…
If $G$ is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs $G_{1}$ and $G_{2}$ is a graph $G=G_{1}\odot G_{2}$ with $V(G)=V(G_{1})\cup…
Inspired by the notion of action convergence in graph limit theory, we introduce a measure-theoretic representation of matrices, and we use it to define a new notion of pseudo-metric on the space of matrices. Moreover, we show that such…
In this paper, we study eigenvalues and eigenfunctions of $p$-Laplacians with Dirichlet boundary condition on graphs. We characterize the first eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By…
We prove a Li-Yau type eigenvalue-diameter estimate for signed graphs. That is, the nonzero eigenvalues of the Laplacian of a non-negatively curved signed graph are lower bounded by $1/D^2$ up to a constant, where $D$ stands for the…
For signed graphs we provide a cubic polynomial upper bound on the multiplicity of its eigenvalues. We show that this bound is sharp by providing examples of signed graphs in which it is attained. We also discuss particular cases in which…
Let $G$ be a unicyclic graph. In this paper, we provide an upper bound for the number of Laplacian eigenvalues of $G$ within the interval $[0,1)$ in terms of the diameter and the girth of $G$.
In this paper, we derive an upper bound for higher order eigenvalues of the normalized Laplace operator associated with a symmetric finite graph in terms of lower order eigenvalues.
We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on…
Consider a semigraph $G=(V,\,E)$; in this paper, we study the eigenvalues of the Laplacian matrix of $G$. We show that the Laplacian of $G$ is positive semi-definite, and $G$ is connected if and only if $\lambda_2 >0.$ Along the similar…
We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.
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…
Graphs possess exotic features like variable size and absence of natural ordering of the nodes that make them difficult to analyze and compare. To circumvent this problem and learn on graphs, graph feature representation is required. A good…
We propose a Laplacian based on general inner product spaces, which we call the inner product Laplacian. We show the combinatorial and normalized graph Laplacians, as well as other Laplacians for hypergraphs and directed graphs, are special…
In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp…