We show that nonconstant eigenfunctions of the p-Laplacian do not necessarily have an average value of 0, as they must when p = 2. This fact has implications for deriving a sharp variational characterization of the second eigenvalue for a general class of nonlinear eigenvalue problems.

Let G=(V,E), $V={v_1,v_2,ldots,v_n}$, be a simple connected graph with $%n$ vertices, $m$ edges and a sequence of vertex degrees $d_1geqd_2geqcdotsgeq d_n>0$, $d_i=d(v_i)$. Let ${A}=(a_{ij})_{ntimes n}$ and ${%D}=mathrm{diag }(d_1,d_2,ldots , d_n)$ be the adjacency and the diagonaldegree matrix of $G$, respectively. Denote by ${mathcal{L}^+}(G)={D}^{-1/2}(D+A) {D}^{-1/2}$ the normalized signles...

We provide an efficient method to calculate the pseudo-inverse of the Laplacian of a bipartite graph, which is based on the pseudo-inverse of the normalized Laplacian.

We give several bounds on the second smallest eigenvalue of the weighted Laplacian matrix of a finite graph and on the second largest eigenvalue of its weighted adjacency matrix. We establish relations between the given Cheeger-type bounds here and the known bounds in the literature. We show that one of our bounds is the best Cheeger-type bound available.

Recently developed numerical methods make possible the highaccuracy computation of eigenmodes of the Laplacian for a variety of “drums” in two dimensions. A number of computed examples are presented together with a discussion of their implications concerning bound and continuum states, isospectrality, symmetry and degeneracy, eigenvalue avoidance, resonance, localization, eigenvalue optimizatio...

We investigate the stability of the multiplicity of the lowest non-zero eigenvalue of the Neumann Laplacian on convex domains in IR under small perturbations.