  In their seminal paper from 1983, Erd\H{o}s and Szemer\'edi showed that any
$n$ distinct integers induce either $n^{1+\epsilon}$ distinct sums of pairs or
that many distinct products, and conjectured a lower bound of $n^{2-o(1)}$.
They further proposed a generalization of this problem, in which the sums and
products are taken along the edges of a given graph $G$ on $n$ labeled
vertices. They conjectured a version of the sum-product theorem for general
graphs that have at least $n^{1+\epsilon}$ edges.
  In this work, we consider sum-product theorems for sparse graphs, and show
that this problem has important consequences already when $G$ is a matching
(i.e., $n/2$ disjoint edges): Any lower bound of the form $n^{1/2+\delta}$ for
its sum-product over the integers implies a lower bound of $n^{1+\delta}$ for
the original Erd\H{o}s-Szemer\'edi problem.
  In contrast, over the reals the minimal sum-product for the matching is
$\Theta(\sqrt{n})$, hence this approach has the potential of achieving lower
bounds specialized to the integers. We proceed to give lower and upper bounds
for this problem in different settings. In addition, we provide tight bounds
for sums along expanders.
  A key element in our proofs is a reduction from the sum-product of a matching
to the maximum number of translates of a set of integers into the perfect
squares. This problem was originally studied by Euler, and we obtain a stronger
form of Euler's result using elliptic curve analysis.
