Some New Results on the Combinatorial Laplacian A THESIS PRESENTED IN PARTIAL FULFILLMENT OF CRITERIA FOR HONORS IN MATHEMATICS
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چکیده
In this thesis we discuss some new results concerning the combinatorial Laplace operator of a simplicial complex. The combinatorial Laplacian of a simplicial complex encodes information about the relationships between adjacent simplices in the complex. This thesis is divided into two relatively disjoint parts. In the first portion of the thesis, we derive a relationship between the Laplacian spectrum of a simplicial complex and the Laplacian spectra of its covering complexes. A covering of a simplicial complex is built from many copies of simplices of the original complex, maintaining the adjacency relationships between simplices. We show that for dimension at least one, the Laplacian spectrum of a simplicial complex is contained inside the Laplacian spectrum of any of its covering
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........................................................................................................ 2 Acknowledgements........................................................................................... 3 Introduction.................................................................................................... 6 Materials and Methods...............................................
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