  In this paper, we use methods from spectral graph theory to obtain some
results on the sum-product problem over finite valuation rings $\mathcal{R}$ of
order $q^r$ which generalize recent results given by Hegyv\'ari and Hennecart
(2013). More precisely, we prove that, for related pairs of two-variable
functions $f(x,y)$ and $g(x,y)$, if $A$ and $B$ are two sets in $\mathcal{R}^*$
with $|A|=|B|=q^\alpha$, then \[\max\left\lbrace |f(A, B)|, |g(A, B)|
\right\rbrace\gtrsim |A|^{1+\Delta(\alpha)},\] for some $\Delta(\alpha)>0$.
