Spectrum of the Laplacian with weights

On the First Curve in the FučiK Spectrum with Weights for a Mixed p-Laplacian

where Δpu = div(|∇u|p−2∇u) is the p-Laplacian of u with 1 < p < ∞, and λ is a real parameter. Moreover, Ω is a smooth bounded domain in RN , N ≥ 1, whose boundary ∂Ω is made of two disjoint nonempty closed sets Γ1 and Γ2 which are smooth manifolds of dimension N − 1, m and n are weights which may be indefinite and unbounded, ∂/∂ν denotes the exterior normal derivative, and u± = max{±u,0}. The m...

A Combinatorial Laplacian with Vertex Weights

One of the classical results in graph theory is the matrix-tree theorem which asserts that the determinant of a cofactor of the combinatorial Laplacian is equal to the number of spanning trees in a graph (see [1, 7, 11, 15]). The usual notion of the combinatorial Laplacian for a graph involves edge weights. Namely, a Laplacian L for G is a matrix with rows and columns indexed by the vertex set ...

Normalized laplacian spectrum of two new types of join graphs

‎Let $G$ be a graph without an isolated vertex‎, ‎the normalized Laplacian matrix $tilde{mathcal{L}}(G)$‎ ‎is defined as $tilde{mathcal{L}}(G)=mathcal{D}^{-frac{1}{2}}mathcal{L}(G)mathcal{D}^{-frac{1}{2}}$‎, where ‎$mathcal{D}$ ‎is a‎ diagonal matrix whose entries are degree of ‎vertices ‎‎of ‎$‎G‎$‎‎. ‎The eigenvalues of‎ $tilde{mathcal{L}}(G)$ are ‎called as ‎the ‎normalized Laplacian eigenva...

On the Fučik Spectrum with Indefinite Weights

The Laplacian Spectrum of Graphs †

The paper is essentially a survey of known results about the spectrum of the Laplacian matrix of graphs with special emphasis on the second smallest Laplacian eigenvalue λ2 and its relation to numerous graph invariants, including connectivity, expanding properties, isoperimetric number, maximum cut, independence number, genus, diameter, mean distance, and bandwidth-type parameters of a graph. S...