Asymptotic spectral distributions of distancek graphs of cartesian product graphs

Let $G$ be a finite connected graph on two or more vertices and $G^{[N,k]}$ the distance-$k$ graph of the $N$-fold cartesian power of $G$. For a fixed $k\ge1$, we obtain explicitly the large $N$ limit of the spectral distribution (the eigenvalue distribution of the adjacency matrix) of $G^{[N,k]}$. The limit distribution is described in terms of the Hermite polynomials.