THESIS
2011
ix, 84 p. : ill. ; 30 cm
Abstract
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 H
12{C
6, (P
1, P
5)} and H
12{C
8, (P
2, P
2)}, 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...[
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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 H
12{C
6, (P
1, P
5)} and H
12{C
8, (P
2, P
2)}, 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.
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