can somebody refer me the article/source in which it has been proved that $G:=K_1\vee (mK_n)$ is determined by its Laplacian spectrum
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Rather Laplacian spectrum of join of any two graphs is completely determined by the Laplacian spectra of the constituting graphs.
R. Merris, Laplacian graph eigenvectors, Linear Algebra and its Applications, 278 (1998), 221--236.
See Theorem 2.1, Page number 225.
G_0_pi_i_e
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in my understanding, this theorem is telling us about the eigenvalue of the join of two graphs,(and he didn`t show all those graphs with same Laplacian spectra must be isomorphic ). – محمد عتیق طا ہر Sep 26 '17 at 03:57
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@محمد عتیق طا ہر Are you asking about spectral characterization? – G_0_pi_i_e Sep 26 '17 at 07:34
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I need to know all those graphs which have same Laplacian eigenvalues as $K_1 \vee (m K_n) $ and they are isomorphic or not. – محمد عتیق طا ہر Sep 26 '17 at 13:05