If an $n$-gon $P$ is isospectral to a regular $n$-gon $Q$, what could we say about the shape of the $P$. Otherwise, what could we say about $Q$? In fact, some hints or simply some ideas should be appreciated.

Clarifications : I talk about the spectrum of the Laplacian on the interior of the polygon, acting on the space of functions vanishing on the boundary.