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In "Spectra of Graphs" by Brouwer and Haemers they gives examples of 2 graphs whose Laplace spectrums are the same, but only one of which is bipartite. I am wondering if there is a similar example for the adjacency spectrum:

1) What's an example of a non-bipartite graph whose adjacency spectrum is still symmetric about 0?

2) Are there graphs $G_1, G_2$ such that they have the same adjacency spectrum but only one of which is bipartite?

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migrated from cs.stackexchange.com Jan 29 '17 at 19:18

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A graph is bipartite iff its adjacency spectrum is symmetric about $0$. See Bipartite Graph. So the bipartiteness of a graph is completely determined by its spectrum and there are no such examples.

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