Figure Largest And The Second Laplace Characteristic Value

Let G be a simple and connected graph with n vertices. The diagonal matrix of vertex degrees and the adjacent matrix of graph G are respectively denoted by D , A . L (G )= D - A is called as the Laplacian matrix of graph G .To research eigenvalues of the Laplacian matrix of graph is important in graph theory and reality., because a lot of invariants are tightly connected with them. bounds of the largest eigenvalue and the second smallest eigenvalue are always utilized in many application. The below problems were solved in regard to bounds ofμ₁ (G )andμn ?1 (G ):1. The main recent results about upper bounds ofμ₁ (G ) was reported and compared with each other.2. The upper bounds of spectral radius of the Laplacian matrices on graphs with accessory theory about bounds of spectral radius of adjacent matrix on digraphs were obtained by utilizing the relations between the row sums of nonnegative matrices and its spectral radius and Perron-Frobenius theory and so on. Meanwhile formulas about the upper bounds of spectral radius were illustrated according to several examples to judge which was the best accuracy and most easily realized by programme among them.3. The upper bounds of spectral radius of Laplacian matrices on Bethe trees, a special graph ,were also researched by us besides them on common simple graphs. The estimation formulas of the upper bounds were obtained by means of characters between matrix theories and trees.4. The second smallest eigenvalue of Laplacian matrices on graph has relations with the algebra connectivity of graph. We also obtained the estimation formulas of bounds in regard to some graphs with cut points or cut edges by detaching L (G )matrix into two matrices of low steps and utilizing the famous Weysl theory. The second smallest eigenvalue of Laplacian matrix of graph with large n was denoted by them of graph with small n .