In this paper, we generalize the combinatorial Laplace operator of Horak and Jost by introducing the @math -weighted coboundary operator induced by a weight function @math . Our weight function @math is a generalization of Dawson's weighted boundary map. We show that our above-mentioned generalizations include new cases that are not covered by previous literature. We outline two applications of @math -weighted (co)homology and the @math -weighted Laplacian in biological sciences. The first application is differentiating ring structures in biomolecules, while the second is distinguishing feedforward loops which are a fundamental type of network motif. We also determine the relation of @math -weighted cohomology and solutions of certain sparse linear systems.
In @cite_27 , J. Hansen and R. Ghrist outlines a program of spectral sheaf theory, which is an extension of spectral graph theory to cellular sheaves. They study weighted cellular sheaves, which is a cellular sheaf with values in @math -vector spaces.
@cite_27
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This paper outlines a program in what one might call spectral sheaf theory --- an extension of spectral graph theory to cellular sheaves. By lifting the combinatorial graph Laplacian to the Hodge Laplacian on a cellular sheaf of vector spaces over a regular cell complex, one can relate spectral data to the sheaf cohomology and cell structure in a manner reminiscent of spectral graph theory. This work gives an exploratory introduction, and includes results on eigenvalue interlacing, complex sparsification, effective resistance, synchronization, and sheaf approximation. These results and subsequent applications are prefaced by an introduction to cellular sheaves and Laplacians. Motivated by S-duality modularity conjectures in string theory, we define new invariants counting a restricted class of two-dimensional torsion sheaves, enumerating pairs (Z H ) in a Calabi–Yau threefold (X ). Here (H ) is a member of a sufficiently positive linear system and (Z ) is a one-dimensional subscheme of it. The associated sheaf is the ideal sheaf of (Z H ), pushed forward to (X ) and considered as a certain Joyce–Song pair in the derived category of (X ). We express these invariants in terms of the MNOP invariants of (X ). We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains. Spectral graph theory is the study of the eigenvalues and eigenvectors of matrices associated with graphs. In this tutorial, we will try to provide some intuition as to why these eigenvectors and eigenvalues have combinatorial significance, and will sitn'ey some of their applications.