Elsevier

Discrete Mathematics

Volume 95, Issues 1–3, 3 December 1991, Pages 193-219
Discrete Mathematics

Some relations between analytic and geometric properties of infinite graphs

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Abstract

For locally finite infinite graphs the following analytic invariants and properties are considered: the spectrum of the transition and difference Laplacian matrix, amenability and the Kazhdan property (T). They are related to several geometric invariants, such as the isoperimetric number, growth, the structure of the space of ends, etc. Usually, only the global behaviour of invariants is important. It is shown how each of the above properties has its ‘essential’ counterpart, e.g. the essential isoperimetric number, the essential spectrum, the essential maximum degree, etc. These invariants do not change if we add or delete finitely many edges in the graph.

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Supported in part by the Research Council of Slovenia, Yugoslavia.