I am revising a paper where one of the operations performed on a undirected graph with no loops, is to take each vertex, and split it into two vertices, and take each edge and replace it with 4 edges: e.g. vertices a and b with edge (a,b) become vertices -a, +a, -b, +b, with edges (-a,-b), (-a,+b), (+a,-b), (+a,+b). A corresponding operation on the adjacency matrix is a Kronecker product with the matrix [[1,1],[1,1]] (the 2 x 2 matrix containing only ones.) The obvious generalization is to replace the 2 x 2 all-ones matrix with an arbitrary symmetric {0,1} matrix.


Are these operations well known? Is there a well known name for them? 



Is there a name for this operation on graphs - e.g. duplication, Kronecker product of graphs?
Its the lexicographic product of the empty graph on two vertices by your original graph. (The terminology for lexicographic product is a bit confused - it's not commutative and there are two schools of thought about which graph should go first.) 


