Spectra Of Signed Graphs And Some Tripartite Graphs

There is a close relation between graph structures and its spectra,usually one can decode one from another.In this paper,we apply graph theory,graph spectra and matrix analysis theory and use fundamental spectra analysis methods,such as Rayleigh-Ritz theorem and the characteristic polynomial of graphs,to study spectra properties of signed graphs and tripartite graphs.In this paper,the specific arrangements are as follows:In the first chapter,we introduce some basic concepts and notations,and then make a brief summary of the research status of graph spectra.Finally,the main conclusions of this paper are listed.In the second chapter,we first introduce some basic concepts of signed graphs and related lemmas.In this chapter,we focus on the study of the balancedness of signed graph.In[2],there are two important conclusions:λn(Γ)≤ε(Γ)andλn(Γ)≤v(Γ),the two conclusions measure the balancedness of signed graph via the minimum Laplacian eigenvalue.In this chapter,we measure the balancedness of signed graph by the minimum normalized Laplacian eigenvalue.Then,we also get this conclusion,i.e the minimum Laplacian eigenvalue of signed graphs can be estimated by the spectral radius of induced subgraph which makes sure the edge-deleted signed subgraph be balanced.In most cases,our conclusion is better than one in[2].In the third chapter,on the basis of circulant graphs and bipartite bicirculant graphs,we continue to extend these results to tripartite graphs.Firstly,we list the preminary knowledge of this chapter;secondly,we give the definition of undirected and directed tripartite graphs;at last,we discuss the connectivity and eigenvalues of these graphs,sufficient condition for the connectedness of them and a method of calculating eigenvalues are presented.In the last chapter,we mainly analyze and discuss the relevant properties of adjacency spec-trum and Laplacian spectrum.And the main content of this chapter are,the second section pro-vides a lower bounds of adjacency spectral radius of signed graphs;The third section gives some new results on the bounds estimation of Laplacian maximum and minimum eigenvalue obtianed in this paper.