Almost self-inverse graphs
Linear and Multilinear Algebra
Let G be a simple graph with adjacency matrix A(G). The inverse of the graph G, when it exists, is the unique weighted graph whose adjacency matrix is similar to A(G)-1 via a particular type of similarity. An invertible graph G is said to be an almost self-inverse graph if the inverse graph of G can be obtained from G just by joining two nonadjacent vertices. We present a characterization of almost self-inverse graphs of a specific class of graphs which contains the class of nonsingular trees.
Bapat, R. B. and Panda, S. K., "Almost self-inverse graphs" (2019). Journal Articles. 661.