On the Spectral Gap of a Quantum Graph
2016 (English)In: Annales de l'Institute Henri Poincare. Physique theorique, ISSN 1424-0637, E-ISSN 1424-0661, 1-35 p.Article in journal (Refereed) PublishedText
We consider the problem of finding universal bounds of “isoperimetric” or “isodiametric” type on the spectral gap of the Laplacian on a metric graph with natural boundary conditions at the vertices, in terms of various analytical and combinatorial properties of the graph: its total length, diameter, number of vertices and number of edges. We investigate which combinations of parameters are necessary to obtain non-trivial upper and lower bounds and obtain a number of sharp estimates in terms of these parameters. We also show that, in contrast to the Laplacian matrix on a combinatorial graph, no bound depending only on the diameter is possible. As a special case of our results on metric graphs, we deduce estimates for the normalised Laplacian matrix on combinatorial graphs which, surprisingly, are sometimes sharper than the ones obtained by purely combinatorial methods in the graph theoretical literature.
Place, publisher, year, edition, pages
Birkhauser Verlag AG , 2016. 1-35 p.
IdentifiersURN: urn:nbn:se:kth:diva-188321DOI: 10.1007/s00023-016-0460-2ScopusID: 2-s2.0-84955282419OAI: oai:DiVA.org:kth-188321DiVA: diva2:934905
QC 201606092016-06-092016-06-092016-06-09Bibliographically approved