Some Graphs Determined By Their Laplacian Spectrum

In recent years, about "what kind of graph is determined by its Laplacian spectrum (that is in a kind of graph,if two graphs have the same Laplacian spectrum, we can obtain that the two graphs are isomorphic, so we say that this kind of graph is determined by its Laplacian spectrum)" has become the research focus.Now, we find a few graphs are determined by their Laplacian spectrum, and prove that most of the graph is not determined by its Laplacian spectrum. But we find that by recombination of the graph which was proved before, we may obtain some new some conclusions. In this article, we use this way constructing the ice cream graph. In addition, the cones over some regular graphs have been proved that they are determined by their signless spectrum, this article further prove that it can also be determined by its Laplacian spectrum.This article’s research method is:We start from the two graphs having the same Laplacian spectrum; according to the lemma, we deduce they have the same polynomial of Laplacian matrix, so the polynomial has the same coefficient; by comparing, we find the relation of vertex degree, then we discuss and obtain the two graphs are isomorphic.In this article, the structure is as follows:In Chapter1, we introduce the development and some results of graph theory and the graph spectra theory, then by introduction some basic concepts, we rise this article. In Chapter2, we give some concepts and the research background of the ice cream graph, then introduce lemma and its proof, we obtain that ice cream graph is determined by its Laplacian spectrum. In Chapter3, we give some concepts and the research background of the cones over some regular graphs, and we prove that the cones over some regular graphs are determined by their Laplacian spectrum.In Chapter4, we give the main conclusion and innovation of this article, also we point out the problem that we can continue to study.