  In the $(G,H)$-isomorphism game, a verifier interacts with two
non-communicating players (called provers) by privately sending each of them a
random vertex from either $G$ or $H$, whose aim is to convince the verifier
that two graphs $G$ and $H$ are isomorphic. In recent work along with Atserias,
\v{S}\'amal and Severini [Journal of Combinatorial Theory, Series B,
136:89--328, 2019] we showed that a verifier can be convinced that two
non-isomorphic graphs are isomorphic, if the provers are allowed to share
quantum resources. In this paper we model classical and quantum graph
isomorphism by linear constraints over certain complicated convex cones, which
we then relax to a pair of tractable convex models (semidefinite programs). Our
main result is a complete algebraic characterization of the corresponding
equivalence relations on graphs in terms of appropriate matrix algebras. Our
techniques are an interesting mix of algebra, combinatorics, optimization, and
quantum information.
