On the spectra of some graphs like weighted rooted trees

Let G be a weighted rooted graph of k levels such that, for j∈{2,…,k}(1)each vertex at level j is adjacent to one vertex at level j-1 and all edges joining a vertex at level j with a vertex at level j-1 have the same weight, where the weight is a positive real number;(2)if two vertices at level j are adjacent then they are adjacent to the same vertex at level j-1 and all edges joining two vertices at level j have the same weight;(3)two vertices at level j have the same degree;(4)there is not a vertex at level j adjacent to others two vertices at the same level;We give a complete characterization of the eigenvalues of the Laplacian matrix and adjacency matrix of G. They are the eigenvalues of leading principal submatrices of two nonnegative symmetric tridiagonal matrices of order k×k and the roots of some polynomials related with the characteristic polynomial of the referred submatrices. By application of the above mentioned results, we derive an upper bound on the largest eigenvalue of a graph defined by a weighted tree and a weighted triangle attached, by one of its vertices, to a pendant vertex of the tree.