  Motivated in part by combinatorial applications to certain sum-product
phenomena, we introduce unimodular graphs over finite fields and, more
generally, over finite valuation rings. We compute the spectrum of the
unimodular graphs, by using Eisenstein sums associated to unramified extensions
of such rings. We derive an estimate for the number of solutions to the
restricted dot product equation $a\cdot b=r$ over a finite valuation ring.
Furthermore, our spectral analysis leads to the exact value of the
isoperimetric constant for half of the unimodular graphs. We also compute the
spectrum of Platonic graphs over finite valuation rings, and products of such
rings - e.g., $\mathbb{Z}/(N)$. In particular, we deduce an improved lower
bound for the isoperimetric constant of the Platonic graph over
$\mathbb{Z}/(N)$.
