  We study a multi-marginal optimal transport problem with surplus $b(x_{1},
\ldots, x_{m})=\sum_{\{i,j\}\in P} x_{i}\cdot x_{j}$, where $P\subseteq
Q:=\{\{i,j\}: i, j \in \{1,2,...m\}, i \neq j\}$. We reformulate this problem
by associating each surplus of this type with a graph with $m$ vertices whose
set of edges is indexed by $P$. We then establish uniqueness and Monge solution
results for two general classes of surplus functions. Among many other
examples, these classes encapsulate the Gangbo and \'{S}wi\c{e}ch surplus [12]
and the surplus $\sum_{i=1}^{m-1}x_{i}\cdot x_{i+1} + x_{m}\cdot x_{1}$ studied
in an earlier work by the present authors [23].
