We study the spectral gap, i.e. the distance between the two lowest eigenvalues for Laplace operators on metric graphs. A universal lower estimate for the spectral gap is proven and it is shown that it is attained if the graph is formed by just one interval. Uniqueness of the minimizer allows to prove a geometric version of the Ambartsumian theorem derived originally for Schrödinger operators.
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Pavel Kurasov, Sergey Naboko, Rayleigh estimates for differential operators on graphs. J. Spectr. Theory 4 (2014), no. 2, pp. 211–219DOI 10.4171/JST/67