A Laplacian matrix, L = (lij) ∈ R , has nonpositive off-diagonal entries and zero row sums. As a matrix associated with a weighted directed graph, it generalizes the Laplacian matrix of an ordinary graph. A standardized Laplacian matrix is a Laplacian matrix with − 1 n ≤ lij ≤ 0 at j 6= i. We study the spectra of Laplacian matrices and relations between Laplacian matrices and stochastic matrice...

RNA secondary structure consists of elements such as stems, bulges, loops. The most obvious and important scalar number that can be attached to an RNA structure is its free energy, with a landscape that governs the folding pathway. However, because of the unique geometry of RNA secondary structure, another interesting single-signed scalar number based on geometrical scales exists that can assis...

There has been recent work [Louis STOC 2015] to analyze the spectral properties of hypergraphs with respect to edge expansion. In particular, a diffusion process is defined on a hypergraph such that within each hyperedge, measure flows from nodes having maximum weighted measure to those having minimum. The diffusion process determines a Laplacian, whose spectral properties are related to the ed...

The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations. In this pape...

We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality f...

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 the structure of ‘local’ subgraphs of the network. We call a subgraph local when it is induced by the set of nodes obtained from a breath-first search (BFS)...

We prove that the first positive eigenvalue, normalized by the volume, of the sub-Laplacian associated with a strictly pseudoconvex pseudo-Hermitian structure θ on the CR sphere S ⊂ C, achieves its maximum when θ is the standard contact form.

In this note we address the problem of graph isomorphism by means of eigenvalue spectra of different matrix representations: the neighborhood matrix M̂ , its corresponding signless Laplacian QM̂ , and the set of higher order adjacency matrices M`s. We find that, in relation to graphs with at most 10 vertices, QM̂ leads to better results than the signless Laplacian Q; besides, when combined with M̂ ...