We present the spectrum of the (normalized) graph Laplacian as a systematic tool for the investigation of networks, and we describe basic properties of eigenvalues and eigenfunctions. Processes of graph formation like motif joining or duplication leave characteristic traces in the spectrum. This can suggest hypotheses about the evolution of a graph representing biological data. To this data, we...

Abstract We consider a convolution-type operator on vector bundles over metric-measure spaces. This extends the analogous convolution Laplacian functions in our earlier work to bundles, and is natural extension of graph connection Laplacian. prove that for Euclidean or Hermitian connections closed Riemannian manifolds, spectrum this both approximate

Abstract. Given two graphs G1, with vertices 1, 2, ..., n and edges e1, e2, ..., em, and G2, the edge corona G1 ⋄G2 of G1 and G2 is defined as the graph obtained by taking m copies of G2 and for each edge ek = ij of G, joining edges between the two end-vertices i, j of ek and each vertex of the k-copy of G2. In this paper, the adjacency spectrum and Laplacian spectrum of G1 ⋄ G2 are given in te...

A graph G is said to be determined by its Q-spectrum if with respect to the signless Laplacian matrix Q , any graph having the same spectrum as G is isomorphic to G. The lollipop graph, denoted by Hn,p, is obtained by appending a cycle Cp to a pendant vertex of a path Pn−p. In this paper, it is proved that all lollipop graphs are determined by their Q -spectra. © 2008 Elsevier B.V. All rights r...

A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, infinitely many non-is...

Abstract. A fractafold, a space that is locally modeled on a specified fractal, is the fractal equivalent of a manifold. For compact fractafolds based on the Sierpiński gasket, it was shown by the first author how to compute the discrete spectrum of the Laplacian in terms of the spectrum of a finite graph Laplacian. A similar problem was solved by the second author for the case of infinite blow...

In this first talk we will introduce three of the most commonly used types of matrices in spectral graph theory. They are the adjacency matrix, the combinatorial Laplacian, and the normalized Laplacian. We also will give some simple examples of how the spectrum can be used for each of these types.

Abstract In this paper, we give the spectrum of a matrix by using the quotient matrix, then we apply this result to various matrices associated to a graph and a digraph, including adjacency matrix, (signless) Laplacian matrix, distance matrix, distance (signless) Laplacian matrix, to obtain some known and new results. Moreover, we propose some problems for further research. AMS Classification: ...

The spectrum of the Laplacian, signless Laplacian and adjacency matrices of the family of the weighted graphs R{H}, obtained from a connected weighted graph R on r vertices and r copies of a modified Bethe tree H by identifying the root of the i-th copy of H with the i-th vertex of R, is determined.