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Eigenvalue estimates on quantum graphs

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arxiv 1609.07471 v1 pith:MQ75QSFR submitted 2016-09-23 math.SP math-phmath.MP

Sinan Ariturk This is my paper · ORCID
classification math.SPmath-phmath.MP
keywords boundsestimateseigenvaluesestablishgraphgraphsnumberpendant
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abstract

On a finite connected metric graph, we establish upper bounds for the eigenvalues of the Laplacian. These bounds depend on the length, the Betti number, and the number of pendant vertices. For trees, these estimates are sharp. We also establish sharp upper bounds for the spectral gap of the complete graph $K_4$. The proofs are based on estimates for eigenvalues on graphs with Dirichlet conditions imposed at the pendant vertices.

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  1. On Courant-type bounds and spectral partitioning via Neumann domains on quantum graphs

    math.SP 2025-09 conditional novelty 5.0 of 10

    On tree quantum graphs, the n-th Laplacian eigenfunction has at most n-1 zeros, and under genericity assumptions the spectral minimal partition energy equals the (n+1)-th Neumann eigenvalue.

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