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I'm working on the adjacency matrix of some graphs and need some facts about Hermitian matrices which have exactly two distinct eigenvalues. Can anybody help me introduce source about spectrum of Hermitian matrices or more generally about these matrices?

Bests.

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  • $\begingroup$ What do you actually want to know about such matrices? Any Hermitian matrix is diagonalizable by the spectral theorem, as found in many linear algebra textbooks, and then you can read off the eigenvalues from the diagonal entries $\endgroup$
    – Yemon Choi
    Commented Jan 14, 2018 at 17:23
  • $\begingroup$ Aren't the adjacency matrices usually symmetric (as opposed to just Hermitian)? $\endgroup$
    – Igor Rivin
    Commented Jan 14, 2018 at 17:39

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It is a well known fact that a graph whose adjacency matrix has two distinct eigenvalues is the complet graph, see

Brouwer, Andries E.; Haemers, Willem H., Spectra of graphs, Universitext. Berlin: Springer (ISBN 978-1-4614-1938-9/hbk; 978-1-4614-1939-6/ebook). xiii, 250 p. (2012). ZBL1231.05001.

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