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I suspect this could be an easy one but I am not an expert in algebraic graph theory.

Let $Q(G)$ define the Laplacian matrix for a simple graph $G$. It is well known that n is an eigenvalue of $Q(K_n)$ with multiplicity $n-1$. I was wondering if graphs $G$ of order $n$ such that $Q(G)$ has an eigenvalue of multiplicity $n-1$ have been characterized.

More specifically, is there any other graph of order $n$ besides $K_n$ such that respective Laplacian matrix has an eigenvalue of multiplicity $n-1$ ?

I think an answer could perhaps be found here but needless to say I do not have access to the paper.

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up vote 4 down vote accepted

Such a graph has only two distinct eigenvalues, and graphs with few eigenvalues have been studied. See

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Just for completeness I will mention, that the author of the thesis states that the only graph with two distinct eigenvalues is the complete graph. – Jernej Jul 11 '11 at 22:02

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