This thesis studies two subjects. One is the spectral characterization problem, the other is the spectral estimation probelm. For the former, we mainly investigate the spectral characterization of graphs H_n{C_q, (P_n1, P_n2)}. It is proved that except for the A-cospectral graphs H₁₂{C₆, (P₁, P₅)} and H₁₂{C₈, (P₂, P₂)}, no two non-isomorphic graphs of the form H_n{C_q, (P_n1, P_n2)} are A-cospectral. And, graph H_n{C_q, (P_n1, P_n2)} is proved to be determined by its L-spectrum. Also, it is proved that all graphs H_n{C_q, (P_n1 , P_n2)} are determined by their Q-spectra, except for graphs H_2a+4{C_a+3, (P_a/2, P_a/2+1)} with a being a positive even number and H_2b{C_b, (P_b/2, P_b/2)} with b ≥ 4 being an even number. For the latter, some spectral estimations of signed graphs are given. We formulate some relati...[ Read more ]

This thesis studies two subjects. One is the spectral characterization problem, the other is the spectral estimation probelm. For the former, we mainly investigate the spectral characterization of graphs H_n{C_q, (P_n1, P_n2)}. It is proved that except for the A-cospectral graphs H₁₂{C₆, (P₁, P₅)} and H₁₂{C₈, (P₂, P₂)}, no two non-isomorphic graphs of the form H_n{C_q, (P_n1, P_n2)} are A-cospectral. And, graph H_n{C_q, (P_n1, P_n2)} is proved to be determined by its L-spectrum. Also, it is proved that all graphs H_n{C_q, (P_n1 , P_n2)} are determined by their Q-spectra, except for graphs H_2a+4{C_a+3, (P_a/2, P_a/2+1)} with a being a positive even number and H_2b{C_b, (P_b/2, P_b/2)} with b ≥ 4 being an even number. For the latter, some spectral estimations of signed graphs are given. We formulate some relations on the eigenvalues of a signed graph. And we find a lower bound on the second largest Laplacian eigenvalue of a signed graph in terms of the largest and the second largest degrees. Also, some upper bounds on the smallest and the second smallest Laplacian eigenvalues of a signed graph in terms of the smallest and the second smallest degrees are formulated.