Abstract
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.
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Communicated by Jan Derezinski.
Part of this work was completed while J. B. K. was the recipient of a fellowship of the Alexander von Humboldt Foundation, Germany. P. K. was partially supported by the Swedish Research Council (Grant D0497301). G. M. and D. M. were partially supported by the Land Baden–Württemberg in the framework of the Juniorprofessorenprogramm—research project on “Symmetry methods in quantum graphs”. All four authors were partially supported by the Center for Interdisciplinary Research (ZiF) in Bielefeld in the framework of the cooperation group on “Discrete and continuous models in the theory of networks”. The authors would like to thank Jens Wirth for helpful discussions regarding Theorem 5.10.
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Kennedy, J.B., Kurasov, P., Malenová, G. et al. On the Spectral Gap of a Quantum Graph. Ann. Henri Poincaré 17, 2439–2473 (2016). https://doi.org/10.1007/s00023-016-0460-2
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DOI: https://doi.org/10.1007/s00023-016-0460-2