Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees

Let G be a simple graph, its Laplacian matrix is the difference of the diagonal matrix of its degrees and its adjacency matrix. Denote its eigenvalues by @m(G)[email protected]"1(G)>[email protected]"2(G)>=...>[email protected]"n(G)=0. A vertex of degree one is called a pendant vertex. Let T"n","k be a tree with n vertices, which is obtained by adding paths P"1,P"2,...,P"k of almost equal the number of its vertices to the pendant vertices of the star K"1","k. In this paper, the following results are given: (1) Let T be a tree with n vertices and k pendant vertices. [email protected](T)= [email protected]?i=1nd"i^2,where equality holds if and only if G is a regular connected bipartite graph. (3) Let G be a simple connected bipartite graph with vertices v"1,v"2,...,v"n and their degrees d"1,d"2,...,d"n. [email protected](G)>[email protected]?v"i~v"j,i