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On the Eigenvalues of Weighted Directed Graphs

Abstract

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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Acknowledgements

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.

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Correspondence to Marwa Balti.

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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

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Keywords

  • Graph Laplacian
  • Bounds of eigenvalues
  • Domain monotonicity
  • Comparison of eigenvalues

Mathematics Subject Classification

  • Primary 47A10
  • 35P15
  • 49R05
  • Secondary 05C50
  • 47A75