Studies On Some Problems Of The Spectral Theory Of Signed Graphs

Algebraic graph theory is an organic combination of graph theory and algebra,and spectral graph theory is an important re-search direction of algebraic graph theory,which mainly studies the spectra of the adjacency matrix,the Laplacian matrix and the the signless Laplacian matrix of a graph.This paper mainly stud-ies the spectral theory of signed graphs,consists of the following parts:1.In Chapter 1,we first introduce the basic concepts of the graph and signed graph,give the research background and status of spectral graph theory;we next introduce the problems to be studied,the background of these problems and list the main results of this thesis.Finally,we will list some basic concepts and notions which are necessary for this thesis.2.In Chapter 2,we will study the sum of the first k Laplacian eigenvalues of a signed graph.In detail,we will prove that the sum of the first two Laplacian eigenvalues of any signed graph is less than or equal to the number of edges plus a constant 4.3.In Chapter 3,we will characterize two classes of signed graphs with few distinct adjacency eigenvalues.In detail,in Section 3.1,we will completely characterize all signed graphs with exactly two adjacency eigenvalues different from ±1.In Section 3.2,we will completely characterize all 5-regular signed graphs with exactly two different adjacency eigenvalues,which solves an open problem raised by Belardo et al.[8].4.In Chapter 4,we mainly study the integral signed graphs.In detail,in Section 4.1,we will completely characterize all adja-cency integral signed graphs of maximum degree 3.In Section 4.2,we will completely characterize all Laplacian integral signed graphs of maximum degree 3.