I have a tree with an invertible adjacency matrix. I know a tree has a invertable matrix if the tree has a unique perfect matching. I take the inverted adjacency matrix of the tree and replace all -1 with a 1, so every such inverse can be understood as an adjacency matrix of a graph. This way I get the inverse of the tree, if I understood correctly. How can I find out which edges the tree and its inverse have in common?
I have found that the edges between the leaves and the neighboring edge also appear in the inverse of the tree. However, there are other edges which are present in both graphs. In the middle of the tree as well as in its inverse without being adjacent to a leaf. How do I recognize these without drawing the inverse of the tree? There must be a certain pattern, i think.