On Graphs With Three Eigenvalues

Let G be a connected undirected simple graph with adjacency matrix A. The eigen-values of G are defined as the eigenvalues of A. The set of numbers which are eigen-values of G, together with their multiplicities is called the spectrum of G.This thesis focus on the following two problems in the theory of graph spectra. The first problem we studied is searching connected nonregular graphs apart from strongly regular graphs and complete bipartite graphs which have just three distinct eigenvalues. And the next problem we studied is that of characterising graphs with second largest eigenvalue at most 1.This thesis is organised as follows:In Section 1 we develop some necessary terminology and preliminaries to under-stand graph and spectral of graph, and introduce the background of our research.In Section 2 we only consider nonregular connected graphs. Firstly we classify nonregular connected graphs that have three distinct eigenvalues and a disconnected complement, and give a series of bound for graphs with exactly three distinct eigen-values, these include the bounds for the number of vetices, valencies, the eigenvalues and the Perron-Frobenius eigenvector. We show that, if a graph and its complements both have precisely 3 distinct eigenvalues, then such a graph have exactly two distinct valencies.Then we continue this investigation focussing mainly on connected graphs with precisely three distinct eigenvalues and exactly two distinct valencies, so called strongly biregular graphs. Our results include structure theorems, the construction of new ex-amples, a classification of certain special families of strongly biregular graphs, and two infinite families of feasible strongly biregular graphs.Finally we contribute a new example of a graph with precisely three distinct eigen-values and exactly three distinct valencies of which there are currently only finitely many known examples. We also show the nonexistence of graphs correponding to cer-tain spectra and valencies.In Section 3 we show a Neumaier-like theorem for strongly biregular graphs, that is, for given positive integer m, the number of connected non-bipartite strongly bireg-ular graphs, and with smallest eigenvalue at least -m or second largest eigenvalue at most m, is finite.In Section 4 we classify the connected graphs with precisely three distinct eigen-values and second largest eigenvalue at most 1, and also classify the connected graphs with precisely three distinct eigenvalues and the smallest valency at most 6 or the largest eigenvalue at most 7.