On The Spectral Radius And The Proposed Laplacian Spectral Radius Upper Bound,

In a simplegraph G=(V,E), Let A(G)=(a_ij)_n×n and D(G)=diag(d₁,d₂,…, d_n) be theadjacency matrix and the diagonal matrix of vertex degrees, respectively. Then L(G)=D(G)-A(G)and Q(G)=D(G)+A(G) are named Laplacian matrix and quasi-Laplacian matrix. The sequenceof all the eigenvalues of A(G), L(G), Q(G) are called the adjacency spectrum, Lalacian spectrum andquasi-Laplacian spectrum, respectively.The graph spectra, t,heory is a new field in graph theory, it originated from the techniques, whichwas first used by theoretic chemists and physicists. In thc past several decades, it has been developedinto a systematic, integrated theory. Literatures and publications on the the adjacency spectrum andLalacian spectrum have been plentiful. On the contrary the study on the quasi-Laplacian spectrum iscomparatively fewer. In fact, it is valuable in application of chemists and physicists.The main results obtained in this thesis can be summarized as follows:1. Introduce relevant concept and term on the graph theory and summary the meaning and someadvance of the study.2. Summary the meaning and some advance of the study on the adjacency spectrum of graphs,and givea new proof of an upper bound on spectral radius of the special 3-connected graph.3. Study the bounds of the quasi-Laplacian spectral radius of Halin graphs and use the similar way tostudy the special 3-connected graph.4. As the similar matrice have the same characteristic polynomial, we determine some sharp upperbounds ofμ₁ of a graph.