The Hecke L-function Hj(s) attached to the jth Maass form for the full modular group is estimated in the mean square over a spectral interval for s = 1 2 + it. As a corollary, we obtain the estimate Hj( 1 2 + it) ¿ t1/3+ε for t À κ j , where 1/4 + κj is the respective jth eigenvalue of the hyperbolic Laplacian. This extends a result due to T. Meurman.

In this paper we give a method to geometrically modify an open set such that the first k eigenvalues of the Dirichlet Laplacian and its perimeter are not increasing, its measure remains constant, and both perimeter and diameter decrease below a certain threshold. The key point of the analysis relies on the properties of the shape subsolutions for the torsion energy.

Here we investigate how the spectrum of the normalized graph Laplacian gets affected by certain graph operations like motif duplication and graph (or motif) joining.

One of the best known results in spectral graph theory is the following lower bound on the chromatic number due to Alan Hoffman, where μ1 and μn are respectively the maximum and minimum eigenvalues of the adjacency matrix: χ ≥ 1+μ1/−μn. We recently generalised this bound to include all eigenvalues of the adjacency matrix. In this paper, we further generalize these results to include all eigenva...

The density matrix of a graph is the combinatorial laplacian matrix of a graph normalized to have unit trace. In this paper we generalize the entanglement properties of mixed density matrices from combinatorial laplacian matrices of graphs discussed in Braunstein et al. [Annals of Combinatorics, 10 (2006) 291] to tripartite states. Then we prove that the degree condition defined in Braunstein e...

Dynamical properties of complex networks are related to the spectral properties of the Laplacian matrix that describes the pattern of connectivity of the network. In particular we compute the synchronization time for different types of networks and different dynamics. We show that the main dependence of the synchronization time is on the smallest nonzero eigenvalue of the Laplacian matrix, in c...

For a connected graph G, we derive tight inequalities relating the smallest signless Laplacian eigenvalue to the largest normalised Laplacian eigenvalue. We investigate how vectors yielding small values of the Rayleigh quotient for the signless Laplacian matrix can be used to identify bipartite subgraphs. Our results are applied to some graphs with degree sequences approximately following a pow...

The heat-kernel of a graph is computed by exponentiating the Laplacian eigen-system with time. In this paper, we study the heat kernel mapping of the nodes of a graph into a vector-space. Specifically, we investigate whether the resulting point distribution can be used for the purposes of graphclustering. Our characterisation is based on the covariance matrix of the point distribution. We explo...

The use of spectral methods in graph theory has allowed for some amazing results where an arithmetic invariant (i.e., diameter, chromatic number, and so on) has been bounded and analyzed using analytic tools. The key has been to examine the spectrum of various matrices associated with graphs and to try to “hear the shape” of the graph from the spectrum. The three most widely used spectrums are ...