On the classification of signed graphs with at most two eigenvalues different from ∓1

Spectral graph theory; thanks to the spectral information obtained from graph matrices, it is a very important sub-branch of graph theory that allows the graph to be changed in a certain way. The problem of spectral characterization of graphs is a problem that has been studied for a long time in the literature. It is seen that many situations made in the graph spectra have been tried to be transferred to the signed graph spectra in recent years. For example, many problems regarding the spectral characterization of graphs have been studied in the literature, such as in [19]. Today, these problems are tried to be transferred to signed graphs. In particular, graphs whose adjacency matrices have at most two eigenvalues different from ±1 have been classified in the literature, and there are studies in the literature [18], [19], [22] on the classification of all signed graphs with a maximum of two eigenvalues different than ±1 . In this thesis, the information and findings about this classification from the literature were examined and the results obtained from these studies were included. In the second chapter, the basic graph theory information about the main sections in the chapters and the algebraic concepts used in the thesis study are presented. Afterwards, all the basic information used about the signed graphs are included in this section. In the third chapter, 'signed graphs with maximum of two eigenvalues different from ±1' are shown with the set G, and graphs with some properties in G (disconnected graph, bipartite graph, complete graph) are presented. In the fourth chapter, the studies carried out to determine all the other graphs except the features in G and in the third chapter are given. In the last section, thoughts on these studies and other studies done so far, as well as suggestions that can be made in the future on this subject are given.