Quantum Graphs which Optimize the Spectral Gap

Ram Band, Guillaume Levy

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)3269-3323
Number of pages55
JournalAnnales Henri Poincare
Issue number10
StatePublished - 1 Oct 2017

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Nuclear and High Energy Physics
  • Mathematical Physics


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