Loading...

Laplacian Eigenvalues of Graphs and their Multiplicity

Fakharan, Mohammad Hossein | 2022

389 Viewed
  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 55158 (02)
  4. University: Sharif University of Technolog
  5. Department: Mathematical Sciences
  6. Advisor(s): Akbari, Saeed
  7. Abstract:
  8. In this thesis, we study the multiplicity of the Laplacian eigenvalues of trees. It is known that for trees, integer Laplacian eigenvalues larger than 1 are simple. Here we consider the multiplicities of the other (non- integral) Laplacian eigenvalues.We provide an upper bound and determine the trees of order n that have a multiplicity that is close to the upper bound (n-3)/2 , and emphasize the particular role of the algebraic connectivity.In continuation, let G be a graph and I be an interval. We present bounds for the number m_G I of Laplacian eigenvalues in I in terms of structural parameters of G. In particular, we show that m_G (n-α(G),n]≤ n-α(G) and m_G (n-d(G)+3,n]≤ n-d(G)-1, where α(G) and d(G) denote the independence number and the diameter of G, respectively. Also, we characterize bipartite graphs that satisfy m_G [0,1)=α(G) Further, in the case of triangle-free or quadrangle-free, we prove that m_G (n-1,n]≤1.Finally, let G be a connected graph of order n with domination number γ(G). Using star sets, in this work we prove that γ(G)≤n-m_G (λ), when γ(G) is an arbitrary adjacency eigenvalue of G, and we characterize the cases of equality. Moreover, we show a connection between start sets and the p-domination number, and we apply it to prove that the aforementioned bound also holds for the p-domination number of a graph γ_p using both the adjacency and Laplacian eigenvalue multiplicities.
  9. Keywords:
  10. Laplacian Matrix ; Laplacian Spectrum ; Adjacency Matrix ; Tree ; Rank ; Laplacian Eigen Values ; P-Pomination Number ; Total Domination Number

 Digital Object List