In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance between two...

We discuss a discrete version of Sunada's Theorem on isospectral manifolds which allows to generate isospectral simple graphs, i.e. nonisomorphic simple graphs which have the same Laplace spectrum. We also consider additional boundary conditions and Buser's transplantation technique applied to a discrete situation.

Spectral graph theory gives an algebraical approach to analyze the dynamics of a network by using the matrix that represents the network structure. However, it is not easy for social networks to apply the spectral graph theory because the matrix elements cannot be given exactly to represent the structure of a social network. The matrix element should be set on the basis of the relationship betw...

Using the Harish-Chandra D-module, we give a proof of Haiman’s theorem on the positivity of Macdonald polynomials. Ginzburg’s work on the connection between this D-module and the isospectral commuting variety is fundamental to this approach.

Video summarization is a powerful tool to handle the huge amount of data generated every day. At shot level, the key-frame extraction problem provides sufficient indexing and browsing of large video databases. In this paper we propose an approach that estimates the number of key-frames using elements of the spectral graph theory. Next, the frames of the video sequence are clustered into groups ...

At some time, in the childhood of spectral graph theory, it was conjectured that non-isomorphic graphs have different spectra, i.e. that graphs are characterized by their spectra. Very quickly this conjecture was refuted and numerous examples and families of non-isomorphic graphs with the same spectrum (cospectral graphs) were found. Still some graphs are characterized by their spectra and seve...

In this paper we explore how a spectral technique suggested by quantum walks can be used to distinguish non-isomorphic cospectral graphs. Reviewing ideas from the field of quantum computing we recall the definition of the unitary matrices inducing quantum walks. We show how the spectra of these matrices are related to the spectra of the transition matrices of classical walks. Despite this relat...