Edge Disjoint Spanning Trees and Eigenvalues

Mehdizadeh, Alireza | 2018

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 51750 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. The spectrum of a graph is related to many important combinatorial parameters. Let (G), ′(G) be the maximum number of edge-disjoint spanning trees and edge-connectivity of a graph G,respectively. Motivated by a question of Seymour on the relationship between eigenvalues of a graph G and bounds of (G), we use eigenvalue interlacing for quotient matrix associated to graph to get the relationship between eigenvalues of a graph and bounds of (G) and ′(G). We also study the relationship between eigenvalues and bounds of (G) and ′(G) in a multigraph G. In the first chapter we prove eigenvalue interlacing and give several applications of it for obtaining bounds for characteristic numbers of graphs, such as the size of a maximal coclique, the chromatic number, the diameter, and the bandwidth, in terms of the eigenvalues of the standard adjacency matrix or the Laplacian matrix.In chapter 2 we mainly discuss edge-connectivity of a graph from spectral perspective. In chapter 3 we give sufficient spectral conditions for existence of k edge-disjoint spanning trees in a graph G. Moreover we prove generalizations of conjecture due to Liu, Hong, and Lai, including some theorems on fractional arboricity and multigraphs
  9. Keywords:
  10. Eigen Values ; Eigenvalue Interlacing ; Laplacian Matrix ; Edge Connectivity ; Edge-Disjoint Spanning Trees ; Quotient Matrix

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