Abstract We consider a convolution-type operator on vector bundles over metric-measure spaces. This extends the analogous convolution Laplacian functions in our earlier work to bundles, and is natural extension of graph connection Laplacian. prove that for Euclidean or Hermitian connections closed Riemannian manifolds, spectrum this both approximate

In this paper, we discuss some properties of relations between a mixed graph and its line graph, which are used to characterize the Laplacian spectrum of a mixed graph. Then we present two sharp upper bounds for the Laplacian spectral radius of a mixed graph in terms of the degrees and the average 2-degrees of vertices and we also determine some extreme graphs which attain these upper bounds. ©...

We consider one dimensional p-Laplacian eigenvalue problems of the form −∆pu = (λ− q)|u|p−1sgnu, on (0, b), together with periodic or separated boundary conditions, where p > 1, ∆p is the p-Laplacian, q ∈ C1[0, b], and b > 0, λ ∈ R. It will be shown that when p 6= 2, the structure of the spectrum in the general periodic case (that is, with q 6= 0 and periodic boundary conditions), can be comple...

We consider the normalized Laplacian matrix for signed graphs and derive interlacing results for its spectrum. In particular, we investigate the effects of several basic graph operations, such as edge removal and addition and vertex contraction, on the Laplacian eigenvalues. We also study vertex replication, whereby a vertex in the graph is duplicated together with its neighboring relations. Th...

We study the time scales associated with diffusion processes that take place on multiplex networks, i.e., on a set of networks linked through interconnected layers. To this end, we propose the construction of a supra-laplacian matrix, which consists of a dimensional lifting of the laplacian matrix of each layer of the multiplex network. We use perturbative analysis to reveal analytically the st...

We prove several results showing that absolutely continuous spectrum for the Laplacian on radial trees is a rare event. In particular, we show that metric trees with unbounded edges have purely singular spectrum and that generically (in the sense of Baire) radial trees have purely singular continuous spectrum.

In this paper, we will introduce the rotation number for the one-dimensional asymmetric p-Laplacian with a pair of periodic potentials. Two applications of this notion will be given. One is a clear characterization of two unbounded sequences of Fučik curves of the periodic Fučik spectrum of the p-Laplacian with potentials. With the help of the Poincaré–Birkhoff fixed point theorem, the other ap...

We discuss the limit of small width for the Laplacian defined on a waveguide with Robin boundary conditions. Under suitable hypothesis on the scaling of the curvature, we prove the convergence of the Robin Laplacian to the Laplacian on the corresponding graph. We show that the projections on each transverse mode generically give rise to decoupling conditions between the edges of the graph while...