This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as the sum of two non self-adjoint Laplacians. We investigate how the perturbation of the graph can affect the eigenvalues. Our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized for graph Laplacians.
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Anné, C., Torki-Hamza, N.: The Gauss-Bonnet operator of an infinite graph. Anal. Math. Phys. 5, 137–159 (2015)
Balti, M.: Non self-adjoint Laplacian on a directed graph (2016, Submitted)
Benson, B.: Sturm-Liouville estimates for the spectrum and Cheeger constant. Int. Math. Res. Not. 2015, 7510–7551 (2015)
Chavel, I.: Eigenvalues in Riemannian Geometry. Academic Press, INC, Boston (1984)
Diestel, R.: Graph theory. Springer-Verlag, New York (2005)
Fiedler, M.: Algebraic Connectivity of Graphs. Czechoslov. Math. J. 23, 298–305 (1973)
Grigor’yan, A.: Analysis on graphs. Lecture Notes, University of Bielefeld, WS (2011/12)
Golinskii, L., Capizzano, S.S.: The asymptotic properties of the spectrum of non symmetrically perturbed Jacobi matrix sequences. J. Approx. Theory 144, 84–102 (2005)
Hasanov, M.: Spectral problems for operator pencils in non-separated root zones. Turk. J. Math 31, 43–52 (2007)
Horn, R.A., Johnson, C.R.: Matrix Analysis, 2nd edn. Cambridge University Press, Cambridge (2013)
Kurasov, P., Malenova, G., Naboko, S.: Spectral gap for quantum graphs and their connectivity. J. Phys. A Math. Theor. 46, 275309 (16pp) (2013)
Pinsky, R.G.: On domain monotonicity for the principal eigenvalues of the Laplacian with a mixed Dirichlet-Neumann boundary condition. Contemp. Math. 387, 245–252 (2005)
Torgasev, A., Petrovic, M.: On the spectrum of infinite graph. Math. Notes 80, 773–785 (2006)
Torki-Hamza, N.: Laplaciens de graphes infinis. I: Graphes métriquement complets. Confluentes Math. 2, 333–350 (2010) [Translated to: Essential self-adjointness for combinatorial Schrödinger operators I- Metrically complete graphs. arXiv:1201.4644v1]
I take this opportunity to express my gratitude to my PhD advisors Colette Anné and Nabila Torki-Hamza for all the fruitful discussions, helpful suggestions and their guidance during this work. This work was financially supported by the “PHC Utique” program of the French Ministry of Foreign Affairs and Ministry of higher education and research and the Tunisian Ministry of higher education and scientific research in the CMCU Project Number 13G1501 “Graphes, Géométrie et théorie Spectrale”. Also I would like to thank the Laboratory of Mathematics Jean Leray of Nantes (LMJL) and the research unity (UR/13ES47) of Faculty of Sciences of Bizerte (University of Carthage) for its financial and its continuous support.
Communicated by Behrndt, Colombo and Naboko.
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Balti, M. On the Eigenvalues of Weighted Directed Graphs. Complex Anal. Oper. Theory 11, 1387–1406 (2017). https://doi.org/10.1007/s11785-016-0615-7
- Graph Laplacian
- Bounds of eigenvalues
- Domain monotonicity
- Comparison of eigenvalues
Mathematics Subject Classification
- Primary 47A10
- Secondary 05C50