Upper eigenvalue bounds for the Kirchhoff Laplacian on embedded metric graphs

We derive upper bounds for the eigenvalues of the Kirchhoff Laplacian on a compact metric graph depending on the graph's genus g. These bounds can be further improved if $g = 0$, i.e. if the metric graph is planar. Our results are based on a spectral correspondence between the Kirchhoff Laplacian and a particular a certain combinatorial weighted Laplacian. In order to take advantage of this correspondence, we also prove new estimates for the eigenvalues of the weighted combinatorial Laplacians that were previously known only in the weighted case.

[1]  Asma Hassannezhad.  Conformal upper bounds for the eigenvalues of the Laplacian and Steklov problem , 2011 .

[2]  James R. Lee,et al.  Metric Uniformization and Spectral Bounds for Graphs , 2010, ArXiv.

[3]  M. Troyanov.  On the moduli space of singular Euclidean surfaces , 2007, math/0702666.

[4]  P. Kuchment,et al.  Introduction to Quantum Graphs , 2012 .

[5]  C. Cattaneo.  The spectrum of the continuous Laplacian on a graph , 1997 .

[6]  A. Kostenko,et al.  Spectral estimates for infinite quantum graphs , 2017, Calculus of Variations and Partial Differential Equations.

[7]  Leonid Friedlander,et al.  Extremal properties of eigenvalues for a metric graph , 2005 .

[8]  Jonathan Rohleder.  Eigenvalue estimates for the Laplacian on a metric tree , 2016, 1602.03864.

[9]  Delio Mugnolo,et al.  On the Spectral Gap of a Quantum Graph , 2015, 1504.01962.

[10]  James R. Lee,et al.  Eigenvalue Bounds, Spectral Partitioning, and Metrical Deformations via Flows , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[11]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[12]  H. Weinberger,et al.  An optimal Poincaré inequality for convex domains , 1960 .

[13]  H. Neidhardt,et al.  Spectral Theory of Infinite Quantum Graphs , 2017, Annales Henri Poincaré.

[14]  E. M. Andreev.  ON CONVEX POLYHEDRA OF FINITE VOLUME IN LOBAČEVSKIĬ SPACE , 1970 .

[15]  E. M. Andreev.  ON CONVEX POLYHEDRA IN LOBAČEVSKIĬ SPACES , 1970 .

[16]  Zhongzhi Zhang,et al.  The normalized Laplacian spectrum of subdivisions of a graph , 2015, Appl. Math. Comput..

[17]  Delio Mugnolo.  Semigroup Methods for Evolution Equations on Networks , 2014 .

[18]  S. Nicaise.  Spectre des réseaux topologiques finis , 1987 .

[19]  R. Band,et al.  Quantum Graphs which Optimize the Spectral Gap , 2016, 1608.00520.

[20]  Shing-Tung Yau,et al.  Eigenvalues of the laplacian of compact Riemann surfaces and minimal submanifolds , 1980 .

[21]  W. Thurston.  The geometry and topology of 3-manifolds , 1979 .

[22]  J. Below.  A characteristic equation associated to an eigenvalue problem on c2-networks , 1985 .

[23]  James E. pLebensohn.  Geometry and the Imagination , 1952 .

[24]  D. Lenz,et al.  Topological Poincar\'e type inequalities and lower bounds on the infimum of the spectrum for graphs , 2018, 1801.09279.

[25]  Nicholas J. Korevaar.  Upper bounds for eigenvalues of conformal metrics , 1993 .

[26]  Delio Mugnolo,et al.  Lower estimates on eigenvalues of quantum graphs , 2019, 1907.13350.

[27]  U. Feige,et al.  Spectral Graph Theory , 2015 .

[28]  Radoslaw K. Wojciechowski,et al.  Cheeger inequalities for unbounded graph Laplacians , 2012, 1209.4911.

[29]  D. Cohen-Steiner,et al.  A transfer principle and applications to eigenvalue estimates for graphs , 2014, 1409.4228.

[30]  Delio Mugnolo.  What is actually a metric graph , 2019, 1912.07549.

[31]  D. Borthwick,et al.  Sharp diameter bound on the spectral gap for quantum graphs , 2019, Proceedings of the American Mathematical Society.

[32]  G. Berkolaiko,et al.  Edge connectivity and the spectral gap of combinatorial and quantum graphs , 2017, 1702.05264.

[33]  C. Kuratowski.  Sur le problème des courbes gauches en Topologie , 1930 .

[34]  Gary L. Miller,et al.  Separators for sphere-packings and nearest neighbor graphs , 1997, JACM.

[35]  Shang-Hua Teng,et al.  Spectral partitioning works: planar graphs and finite element meshes , 1996, Proceedings of 37th Conference on Foundations of Computer Science.

[36]  Jonathan A. Kelner.  Spectral partitioning, eigenvalue bounds, and circle packings for graphs of bounded genus , 2004, STOC '04.