Connected Graph Of The Spectral Radius And Pull Spectrum Las Spectral Radius Estimates

Let G be a simple and connected graph with n vertices, m edges and denote A(G) to be adjacency matrix of G . If vertices u and v are connected, we denote au v to be vertices u and v edge. We assume thatμ1≥μ2≥μn are the eigenvalues of A( G ).μ1 is regarded as the largest eigenvalue of A( G ), denotedρ(G ).For any u∈V, the degree of u is denoted by d u. Let { }D = D (G ) = diag d1 , d 2, , dnbe the diagonal matrix of vertex degree. The Laplacian matrix of G is L (G ) = D (G ) ? A( G ). From this fact and Gersgorin's theorem, it follows that its eigenvalues are nonnegative real numbers. Since its rows sum to 0, 0 is the smallest eigenvalues for L (G ). We assume thatλ1 (G )≥λ2 (G )≥≥λn ?1(G )≥λn(G ) = 0 are the eigenvalues for L (G ),λ1 (G )is regarded as the largest eigenvalue of L (G ). There are some known Laplacian spectral radius ofλL ( G). K (G ) = D (G ) + A( G ) is called the Qusi-Laplacian matrix of G , Qusi-Laplacian matrix of G is a symmetric and irreducible nonnegative matrix. Adjacency matrix of G and Laplacian matrix of G spectral radius are important in graph theory, because they have a close relation to numerous graph invariants. In many applications, Adjacency matrix of G and Laplacian matrix of G spectral radius are needed. In this paper, Adjacency matrix of G and Laplacian matrix of G spectral radius'results are obtained.(1)In this paper, by using algebraic techniques, this paper presents the upper bound ofρ(G )and the extreme graphs arriving the bound, thus we improve the some results.(2)Another type of upper bound forλL ( G) in terms of the degree sequence and the edge number of G are obtained by utilizing the inequality techniques and relation of matrix B with its eigenvalues, and we determine the extreual graphs.(3)As the similar matrice have the same eigenvalues, we determine some new sharp upper bounds ofλL ( G)of simple connected bipartite graphs.