In mathematics, the Perkel graph, named after Manley Perkel, is a 6-regular graph with 57 vertices and 171 edges.

It is the unique distance-regular graph with intersection array (6, 5, 2; 1, 1, 3).

[1] The Perkel graph is also distance-transitive.

It is also the skeleton of an abstract regular polytope, the 57-cell.

The vertex set is Z3 × Z19 where (i,j) is joined to (i+1,k) when (k-j)3 = 26i.