A matchings dual for graphs I
David Alexander Gregory · 2011
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 S 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 S 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 S may depend on S, it is always a subgraph of G ∗ and is equal to G ∗ for some choice of S with integer