Spectral gap for quantum graphs and their edge connectivity

The spectral gap for Laplace operators on metric graphs and the relation between the graph’s edge connectivity is investigated, in particular what happens to the gap if an edge is added to (or deleted from) a graph. It is shown that, in contrast to discrete graphs, the 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.

• Publication year

  2013

• Venue

  Journal of Physics A: Mathematical and Theoretical

• Publication date

  2013-02-20

• Fields of study

  Mathematics, Physics

• Identifiers
  DOI 10.1088/1751-8113/46/27/275309arXiv 1302.5022
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar

• No claims are published for this paper.

• No concepts are published for this paper.

• Decaying Turbulence in the Generalised Burgers Equation

  2012cited by this paper
• Spectra of Graphs

  2011cited by this paper
• On the ground state of quantum graphs with attractive δ-coupling

  2011influential reference
• Dependence of the spectrum of a quantum graph on vertex conditions and edge lengths

  2010cited by this paper
• Graph Laplacians and topology

  2008cited by this paper
• Schrödinger operators on graphs and geometry I: Essentially bounded potentials

  2008cited by this paper
• Index theorems for quantum graphs

  2007cited by this paper
• Quantum Graphs and Their Applications

  2006cited by this paper
• Extremal properties of eigenvalues for a metric graph

  2005cited by this paper
• Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs

  2004cited by this paper
• Quantum graphs: I. Some basic structures

  2004cited by this paper
• Quantum graphs

  2004cited by this paper
• The Laplacian spectrum of graphs

  2001cited by this paper
• Kirchhoff's rule for quantum wires

  1998cited by this paper
• Spectres de graphes

  1998cited by this paper
• The spectrum of the continuous Laplacian on a graph

  1997cited by this paper
• Spectra of Graphs: Theory and Applications

  1997cited by this paper
• Spectral Graph Theory

  1996cited by this paper
• An Introduction to the Bootstrap

  1995cited by this paper
• Spectra of graphs : theory and application

  1995cited by this paper
• THE LAPLACIAN SPECTRUM OF GRAPHS y

  1991cited by this paper
• Free quantum motion on a branching graph

  1989cited by this paper
• Scattering problems on noncompact graphs

  1988cited by this paper
• Spectra of graphs

  1980cited by this paper
• Algebraic connectivity of graphs

  1973cited by this paper

• Probing the influence of topological and geometric disorder on the spectrum of the differential Laplacian operator on networks

  2026cites this paper
• Optimization of Quantum Graph Eigenvalues with Preferred Orientation Vertex Conditions

  2025cites this paper
• On the asymptotic behavior of the spectral gap for discrete Schr\"odinger operators

  2025cites this paper
• Maximal dissipative operators on metric graphs: Real eigenvalues and their multiplicities

  2025cites this paper
• Isospectral graphs via inner symmetries

  2024cites this paper
• Optimizing the fundamental eigenvalue gap of quantum graphs

  2024cites this paper
• Optimization of quantum graph eigenvalues with preferred orientation vertex conditions

  2024cites this paper
• Geometric spectral theory of quantum graphs

  2023cites this paper
• Mean Distance on Metric Graphs

  2023cites this paper
• Gaps between consecutive eigenvalues for compact metric graphs

  2023cites this paper
• Surgery Transformations and Spectral Estimates of δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{docum

  2022cites this paper
• Spectral gaps for star-like quantum graph and for two coupled rings

  2022cites this paper
• Surgery transformations and eigenvalue estimates for quantum graphs with $\delta'$ vertex interactions

  2022cites this paper
• On isospectral metric graphs

  2021cites this paper
• On the Spectral Gap of a Discrete Schrödinger Operator on the Path Graph in the Limit of Large Volume

  2021cites this paper
• n-Laplacians on Metric Graphs and Almost Periodic Functions: I

  2020cites this paper
• A Family of Diameter-Based Eigenvalue Bounds for Quantum Graphs

  2020cites this paper
• On the hot spots of quantum graphs

  2020cites this paper
• A theory of spectral partitions of metric graphs

  2020cites this paper
• Quantum trees which maximize higher eigenvalues are unbalanced

  2020cites this paper
• The Krein–von Neumann Extension for Schrödinger Operators on Metric Graphs

  2020cites this paper
• GLUING GRAPHS AND SPECTRAL GAP : TITCHMARSH-WEYL OPERATOR-FUNCTION APPROACH

  2019cites this paper
• Optimal Potentials for Quantum Graphs

  2019cites this paper
• Edge switching transformations of quantum graphs—a scattering approach

  2019cites this paper
• Sharp diameter bound on the spectral gap for quantum graphs

  2019cites this paper
• Laplacians on bipartite metric graphs

  2019cites this paper
• Spectral gap of the discrete Laplacian on triangulations

  2019cites this paper
• Optimal Potentials for Quantum Graphs

  2019cites this paper
• Spectral Monotonicity for Schrödinger Operators on Metric Graphs

  2018cites this paper
• On the hot spots of quantum trees

  2018cites this paper
• A sharp eigenvalue bound for quantum graphs in terms of their diameter

  2018cites this paper
• Surgery principles for the spectral analysis of quantum graphs

  2018cites this paper
• Limits of quantum graph operators with shrinking edges

  2018cites this paper
• On the Sharpness of Spectral Estimates for Graph Laplacians

  2018cites this paper
• Edge switching transformations of quantum graphs

  2017cites this paper
• Edge connectivity and the spectral gap of combinatorial and quantum graphs

  2017cites this paper
• Surgery of graphs and spectral gap: Titchmarsh-Weyl operator-function approach

  2017cites this paper
• Eigenvalues and eigenvectors of a system of Bernoulli Euler beams connected together in a tree topology

  2017cites this paper
• Surgery of Graphs: M-Function and Spectral Gap

  2017cites this paper
• Laplaciens non auto-adjoints sur un graphe orienté.

  2017influential citation
• Clustering for metric graphs using the p-Laplacian

  2016cites this paper
• On the Eigenvalues of Weighted Directed Graphs

  2016influential citation
• Quantum Graphs which Optimize the Spectral Gap

  2016cites this paper
• Eigenvalue estimates on quantum graphs

  2016cites this paper
• Rényi and Tsallis entropies related to eigenfunctions of quantum graphs

  2016cites this paper
• Eigenvalue estimates for the Laplacian on a metric tree

  2016influential citation
• Schrödinger operators on graphs: Symmetrization and Eulerian cycles

  2015cites this paper
• On the Spectral Gap of a Quantum Graph

  2015influential citation
• ISOSPECTRALITY FOR GRAPH LAPLACIANS UNDER THE CHANGE OF COUPLING AT GRAPH VERTICES: NECESSARY AND SUFFICIENT CONDITIONS

  2014cites this paper
• Special vector configurations in geometry and integrable systems

  2014cites this paper
• Evolution Equations Associated with Self-Adjoint Operators

  2014cites this paper
• The first eigenvalue of the $$p-$$p-Laplacian on quantum graphs

  2014cites this paper
• Semigroup Methods for Evolution Equations on Networks

  2014cites this paper
• Isospectrality for graph Laplacians under the change of coupling at graph vertices

  2014cites this paper
• On the Spectral Gap for Laplacians on Metric Graphs

  2013cites this paper
• THE SPECTRUM OF THE HILBERT SPACE VALUED SECOND DERIVATIVE WITH GENERAL SELF-ADJOINT BOUNDARY CONDITIONS

  2012cites this paper