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 <https://springerlink3.metapress.com/content/a01321p632887837/resource-secured/?target=fulltext.pdf&sid=rrxkjv553gcfwriiuwfaip55&sh=www.springerlink.com> but needless to say I do not have access to the paper.



Graphs of order n with a Laplacian eigenvalue of multiplicity n-1.
Such a graph has only two distinct eigenvalues, and graphs with few eigenvalues have been studied. See


<http://cage.ugent.be/geometry/Theses/30/evandam.pdf>


