Spectral gap for quantum graphs and their connectivity

Abstract

The spectral gap for Laplace operators on metric graphs is investigated in relation to graph's connectivity, in particular what happens if an edge is added to (or deleted from) a graph. It is shown that in contrast to discrete graphs connection between the connectivity and the spectral gap is not one-to-one. The size of the spectral gap depends not only on the topology of the metric graph but on its geometric properties as well. It is shown that adding sufficiently large edges as well as cutting away sufficiently small edges leads to a decrease of the spectral gap. Corresponding explicit criteria are given.

19 pages, 4 figures

References in corpus (5)

• Kirchhoff's Rule for Quantum Wires
• Quantum Graphs II: Some spectral properties of quantum and combinatorial graphs
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• Index theorems for quantum graphs
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• The first eigenvalue of the $p-$Laplacian on quantum graphs
• On the eigenvalues of weighted directed graphs
• Eigenvalue estimates on quantum graphs
• Edge switching transformations of quantum graphs
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs
• Isospectrality for graph Laplacians under the change of coupling at graph vertices: necessary and sufficient conditions
• Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap
• Spectral Gap of The Discrete Laplacian On Triangulations
• On the spectral gap of the path graph in the limit of large volume
• Geometric spectral theory of quantum graphs