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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.
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ScienceDirect