# Graphs of order n with a Laplacian eigenvalue of multiplicity n-1.

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 $n-1$ ?