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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... 

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...