Laplacian Spectral Characterization Of Some Graphs With Circles

The spectral determination theory of graph is a new field in graph theory, which mainly concerns the adjacency spectrum, the Laplacian spectrum and the signless laplacian spectrum. The question "which graphs are determined by their spectra?" originated from chemistry about half a century ago. At present, results about the question are still limited, and all graphs found to be determined by their spectrum have very special structure. There is no good tool to treat general graphs concerning this problem.A graph G is determined by its spectrum if any graph with the same spectrum is isomorphic to it. The main work of this thesis includes:some results related to the Laplacian spectra of the 4-rose graph (Fig.3-1), spectral determination of the lollipop graph with q pendant edges (Fig.4-2,4-3) and the oo-graph with q pendant edges (Fig.4-7). In addition, we also prove that two new families of special graphs are determined by their Laplacian spectrum.The thesis is divided into the following five parts. The first part introduces the research background and development of the spectral determination problem;In the second part, we present some basic definitions and lemmas in the spectral theory of graphs;The third part studies basic properties of the Laplacian spectrum of 4-rose graphs and proves that two families of those graphs have different Laplacian spectrum;In the fourth part, Laplacian spectral determination is established for H(n,p, q) with p even and H(p+2, p, q) with p odd. It is also proved that U(r, s, t) is de-termined by its Laplacian spectrum;The last part includes the proof of the Laplacian spectral determination of two more families of graphs,θ(l,l,l) and (Cr∩Pt)▽Km.