Quantum graphs which optimize the spectral gap

Abstract

A finite discrete graph is turned into a quantum (metric) graph once a finite length is assigned to each edge and the one-dimensional Laplacian is taken to be the operator. We study the dependence of the spectral gap (the first positive Laplacian eigenvalue) on the choice of edge lengths. In particular, starting from a certain discrete graph, we seek the quantum graph for which an optimal (either maximal or minimal) spectral gap is obtained. We fully solve the minimization problem for all graphs. We develop tools for investigating the maximization problem and solve it for some families of graphs.

References in corpus (3)

• On the Spectral Gap of a Quantum Graph
• Edge connectivity and the spectral gap of combinatorial and quantum graphs
• Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs

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• On the Decomposition of the Laplacian on Metric Graphs
• Interlacing and Friedlander-type inequalities for spectral minimal partitions of metric graphs
• Impediments to diffusion in quantum graphs: geometry-based upper bounds on the spectral gap
• On the spectral gap of the path graph in the limit of large volume