Volume growth and spectrum for general graph Laplacians

Details Volume growth and spectrum for general graph Laplacians Volume growth and spectrum for general graph Laplacians

2012-04-21

arxiv.org/abs/1204.4770v1

Mathematics

Functional Analysis

Scientific paper

We prove estimates relating exponential or sub-exponential volume growth of weighted graphs to the bottom of the essential spectrum for general graph Laplacians. The volume growth is computed with respect to a metric adapted to the Laplacian, and use of these metrics produces better results than those obtained from consideration of the graph metric only. Conditions for absence of the essential spectrum are also discussed. These estimates are shown to be optimal or near-optimal in certain settings and apply even if the Laplacian under consideration is an unbounded operator.

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