A locally \(nK_2\) graph \(G\) is a graph such that the set of neighbors of any vertex of \(G\) induces a subgraph isomorphic to \(nK_2\). We show that a locally \(nK_2\) graph \(G\) must have at least \(6n – 3\) vertices, and that a locally \(nK_2\) graph with \(6n – 3\) vertices exists if and only if \(n \in \{1, 2, 3, 5\}\), and in these cases the graph is unique up to isomorphism. The case \(n = 5\) is surprisingly connected to a classic theorem of algebraic geometry: The only locally \(5K_2\) graph on \(6 \times 5 – 3 = 27\) vertices is the incidence graph of the 27 straight lines on any nonsingular complex projective cubic surface.