In this paper, we use methods from spectral graph theory to obtain some results on the sum–product problem over finite valuation rings R of order q r which generalize recent results given by Hegyvári 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 R ∗ with | A | = | B | = q α , then max { | f ( A , B ) | , | g ( A , B ) | } ≫ | A | 1 + Δ ( α ) , for some Δ ( α ) > 0 . [ABSTRACT FROM AUTHOR]