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

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 image. A vertex of degree one is called a pendant vertex. Let image be a tree with n vertices, which is obtained by adding paths image of almost equal the number of its vertices to the pendant vertices of the star image. In this paper, the following results are given:μ(T)⩽μ(Tn,k),μ(G)⩾21n∑i=1ndi2,μ(G)⩾2+1m∑vi∼vj,i