Related papers: Mysteries around the graph Laplacian eigenvalue 4
We study both numerically and analytically what happens to a random graph of average connectivity "alpha" when its leaves and their neighbors are removed iteratively up to the point when no leaf remains. The remnant is made of isolated…
Here we study the spectral properties of an underlying weighted graph of a non-uniform hypergraph by introducing different connectivity matrices, such as adjacency, Laplacian and normalized Laplacian matrices. We show that different…
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.
We show that by restricting the degrees of the vertices of a graph to an arbitrary set \( \Delta \), the threshold point $ \alpha(\Delta) $ of the phase transition for a random graph with $ n $ vertices and $ m = \alpha(\Delta) n $ edges…
In classic graph signal processing, given a real-valued graph signal, its graph Fourier transform is typically defined as the series of inner products between the signal and each eigenvector of the graph Laplacian. Unfortunately, this…
Based on matrix perturbation theory, closed-form analytic expansions are studied for a Laplacian eigenvalue of an undirected, possibly weighted graph, which is close to a unique degree in that graph. An approximation is presented to provide…
In this paper, it is shown that the graph $T_4(p,q,r)$ is determined by its Laplacian spectrum and there are no two non-isomorphic such graphs which are cospectral with respect to adjacency spectrum.
The eigenvalues of the normalized Laplacian matrix of a network plays an important role in its structural and dynamical aspects associated with the network. In this paper, we study the spectra and their applications of normalized Laplacian…
We consider a model for random hypergraphs with identifiability, an analogue of connectedness. This model has a phase transition in the proportion of identifiable vertices when the underlying random graph becomes critical. The phase…
Using our previously published algorithm, we analyze the eigenvectors of the generalized Laplacian for two metric graphs occurring in practical applications. As expected, localization of an eigenvector is rare and the network should be…
We investigate the distribution of eigenvalues of weighted adjacency matrices from a specific ensemble of random graphs. We distribute $N$ vertices across a fixed number $\kappa$ of components, with asymptotically $\alpha_j \dot N$ vertices…
Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…
The signless Laplacian matrix of a graph $G$ is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called $Q$-eigenvalues of $G$. A $Q$-eigenvalue of a graph $G$ is called a $Q$-main eigenvalue…
We show that, with very high probability, the random graph Laplacian has simple spectrum. Our method provides a quantitatively effective estimate of the spectral gaps. Along the way, we establish results on affine no-gaps delocalization,…
The principal ratio of a graph is the ratio of the greatest and least entry of its principal eigenvector. Since the principal ratio compares the extreme values of the principal eigenvector it is sensitive to outliers. This can be…
For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular…
The Laplacian eigenvalues of a network play an important role in the analysis of many structural and dynamical network problems. In this paper, we study the relationship between the eigenvalue spectrum of the normalized Laplacian matrix and…
We are concerned with spectral problems of the Goldberg-Coxeter construction for $3$- and $4$-valent finite graphs. The Goldberg-Coxeter constructions $\mathrm{GC}_{k,l}(X)$ of a finite $3$- or $4$-valent graph $X$ are considered as…
We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which…
The main result of this paper states that in a rooted product of a path with rooted graphs which are disposed in a somewhat mirror-symmetric fashion, there are distinct eigenvalues supported in the end vertices of the path which are too…