On the spectral gap of a quantum graph
2015 (English)Report (Other academic)
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
2015. , 26 p.
Research Reports in Mathematics, ISSN 1401-5617 ; 5
Quantum graphs, Sturm-Liouville problems, Bounds on spectral gaps
Research subject Mathematics
IdentifiersURN: urn:nbn:se:su:diva-116835OAI: oai:DiVA.org:su-116835DiVA: diva2:808657