To some extent, graph evolutionary mechanisms can be explained by its spectra. Here, we are interested in two graph operations, namely, motif (subgraph) doubling and attachment that are biologically relevant. We investigate how these two processes affect the spectrum of the normalized graph Laplacian. A high (algebraic) multiplicity of the eigenvalues 1, 1±0.5, 1± √ 0.5 and others has been obse...

We consider the p–Laplacian operator on a domain equipped with a Finsler metric. After deriving and recalling relevant properties of its first eigenfunction for p > 1, we investigate the limit problem as p → 1.

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, Rσ(z) := ∑ k (z − λk) σ +. Here {λk} ∞ k=1 are the ordered eigenvalues of the Laplacian on a bounded domain Ω ⊂ Rd, and x+ := max(0, x) denotes the positive part of the quantity x. As corollaries of these inequalities, we derive Weyl-type bounds on λk, on averages such as λ...

A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph G with diameter k and vertex set V , the smallest nontrivial eigenvalue λ1 of the normalized Laplacian L satisfies λ1 6 1− σ ( 1− c k ) for some constant c where σ = 2 ∑ v dv √ d...

We first establish the relationship between the largest eigenvalue of the Laplacian matrix of a graph and its bipartite density. Then we present lower and upper bounds for the largest Laplacian eigenvalue of a graph in terms of its largest degree and diameter.

We consider the minimization problem min Ω∈X (Λ2 − Λ∞) (Ω), where Λ2(Ω) and Λ∞(Ω) are the (square root of the) first eigenvalue of the Laplacian and the first eigenvalue of the ∞−Laplacian respectively. X is the class of convex domains with prescribed diameter. We prove existence of a solution, and we provide several geometrical properties of minimizers.

We present a new motion segmentation algorithm: the Enhanced Local Subspace Affinity (ELSA). Unlike Local Subspace Affinity, ELSA is robust in a variety of conditions even without manual tuning of its parameters. This result is achieved thanks to two improvements. The first is a new model selection technique for the estimation of the trajectory matrix rank. The second is an estimation of the nu...

A basic eigenvalue bound due to Alon and Boppana holds only for regular graphs. In this paper we give a generalized Alon-Boppana bound for eigenvalues of graphs that are not required to be regular. We show that a graph G with diameter k and vertex set V , the smallest nontrivial eigenvalue λ1 of the normalized Laplacian L satisfies λ1 6 1− σ ( 1− c k ) for some constant c where σ = 2 ∑ v dv √ d...

We propose a new formulation called hyperedge expansion (HE) for hypergraph learning. The HE expansion transforms the hypergraph into a directed graph on the hyperedge level. Compared to the existing works (e.g. star expansion or normalized hypergraph cut), the learning results with HE expansion would be less sensitive to the vertex distribution among clusters, especially in the case that clust...