A matchings dual for graphs

Let G be a graph with adjacency matrix A and let F be a field. An F-matrix Q is a support matrix of G if A = [Q ̸=O], the zero-nonzero pattern of Q. If G has an invertible skew-symmetric support F-matrix S, the S-dual G of G is defined as the graph with adjacency matrix [S−1 ̸= O]. An analogous adjacency matrix dual, G has been examined in the literature for those bipartite graphs G with unique perfect matchings for which A−1 is sign-similar to an adjacency matrix. For such graphs G, the +-dual is a an example of an S-dual, that is, G ∼= G for some choice of S. If G is a graph with a perfect matching, the matchings dual of G is the graph G∗ on the same vertex set but with vertices i, j adjacent in G∗ if and only if G− i− j has a perfect matching. Though G may depend on S, it is always a subgraph of G∗ and is equal to G∗ for some choice of S with integer entries. For a large class of graphs, the corona graphs, it turns out that G∗ ∼= G whenever G is defined.