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.