Geometric monotone properties of the first nonzero eigenvalue of Laplacian form operator under the action of the Ricci flow in a compact nmanifold ( ) 2 ≥ n are studied. We introduce certain energy functional which proves to be monotonically non-decreasing, as an application, we show that all steady breathers are gradient steady solitons, which are Ricci flat metric. The results are also extend...

Recently the signless Laplacian matrix of graphs has been intensively investigated. While there are many results about the largest eigenvalue of the signless Laplacian, the properties of its smallest eigenvalue are less well studied. The present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless Laplacian.

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...

The independence number, coloring number and related parameters are investigated in the setting of oriented hypergraphs using spectrum normalized Laplace operator. For both an inertia--like bound a ratio--like shown. A Sandwich Theorem involving clique vector chromatic is proved, as well lower for terms smallest largest eigenvalue Laplacian. In addition, spectral partition numbers studied relat...

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...

The Laplacian spread of a graph is defined to be the difference between the largest eigenvalue and the second smallest eigenvalue of the Laplacian matrix of the graph. In this paper, we investigate Laplacian spread of graphs, and prove that there exist exactly five types of tricyclic graphs with maximum Laplacian spread among all tricyclic graphs of fixed order.

In this paper, we study large m asymptotics of the l1 minimal m-partition problem for Dirichlet eigenvalue. For any smooth domain Ω ⊂ ℝn such that ∣Ω∣ = 1, prove limit $${\rm{lim}}_{m \to \infty}l_m^1\left(\Omega \right) {c_0}$$ exists, and constant c0 is independent shape Ω. Here, $$l_m^1\left({\rm{\Omega}} \right)$$ denotes value normalized sum first Laplacian eigenvalues