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The Laplacian Spectrum of Graphs

Qazizadeh, Majideh | 2009

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 39398 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. In this thesis we investigate the spectrum of the Laplacian matrix of a graph. Although its use dates back to Kirchhoff, most of the major results are much more recent. The first chapter of this thesis is devoted to the integral Laplacian eigenvalues of graphs. In Section 2, particular attention is given to multiplicities of integer eigenvalues and to the effect on the spectrum of various modifications. In Section 3, the Laplacian integral graphs are investigated. The Section 4 relates the degree sequence and the Laplacian spectrum through majorization.The second chapter presents the result on permanent of the Laplacian matrix of graphs and permanental roots. In Section 2, we investigate bounds on the Laplacian permanent for trees which depend only on the number of vertices, the size of a matching, and the length of a path. Since a connected bipartite graph has a spanning tree, the lower bounds will hold also for connected bipartite graphs. In addition we investigate bounds for the vertices in each set of its bipartition. In Section 3, we investigate bounds for the ratio of the Laplacian permanent of a graph to the product of the degrees of the vertices. The final section is devoted to the relationship between pemanental roots and the star degree of a graph
  9. Keywords:
  10. Tree ; Laplacian Eigen Values ; Laplacian Integral Graphs ; Star Degree

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