| The graph spectra theory is a new field in graph theory.It originated from techniques,whichwas first used by theoretic chemists and physicists,of seeking approximate numerical solution for certain partial differential equations.The question "which graphs are determined by their spectrum?" originated from chemistry half a century ago.In 1956,G(u|¨)nthard and Primas raised the question in a paper that relates the theory of graph Spectra to H(u|¨)ckel's theory from chemistry.Another application comes from Fisher in 1966.He modelled the shape of the drum by a graph.Then the sound of that drum is characterised by the eigenvalues of the graph.This question is essentially ours.Almost all trees cannot be determined by their adjacency spectrum or Laplacian spectrum.Answering the question for adjacency or Laplacian matrices seems out of research.In the past half century,there are many results about the DS graphs and non-DS graphs.In this paper,we prove that:(1) The F-shape tree is determined by its Laplacian spectrum.(2) Graph M is determined by its Laplacian spectrum.(3) The上-shape tree H(l,m,n) is determined by its haplacian spectrum.(4) The h-shape tree h(l,m,n) is determined by its Laplacian spectrum.(5) TheⅡ-shape tree is determined by its Laplacian spectrum.(6) Tree Y_n is determined by its Laplacian spectrum.(7) Tree Z_n is determined by its Laplacian spectrum. |
