Inverting non-invertible trees

If a graph has a non-singular adjacency matrix, then one may use the inverse matrix to define a (labeled) graph that may be considered to be the inverse graph to the original one. It has been known that an adjacency matrix of a tree is non-singular if and only if the tree has a unique perfect matching; in this case the determinant of the matrix turns out to be $\pm 1$ and the inverse of the tree was shown to be `switching-equivalent' to a simple graph [C. Godsil, Inverses of Trees, Combinatorica 5 (1985), 33--39]. Using generalized inverses of symmetric matrices (that coincide with Moore-Penrose, Drazin, and group inverses in the symmetric case) we prove a formula for determining a `generalized inverse' of a tree.Comment: 14 page

On a Construction of Integrally Invertible Graphs and their Spectral Properties

Godsil (1985) defined a graph to be invertible if it has a non-singular adjacency matrix whose inverse is diagonally similar to a nonnegative integral matrix; the graph defined by the last matrix is then the inverse of the original graph. In this paper we call such graphs positively invertible and introduce a new concept of a negatively invertible graph by replacing the adjective `nonnegative' by `nonpositive in Godsil's definition; the graph defined by the negative of the resulting matrix is then the negative inverse of the original graph. We propose new constructions of integrally invertible graphs (those with non-singular adjacency matrix whose inverse is integral) based on an operation of `bridging' a pair of integrally invertible graphs over subsets of their vertices, with sufficient conditions for their positive and negative invertibility. We also analyze spectral properties of graphs arising from bridging and derive lower bounds for their least positive eigenvalue. As an illustration we present a census of graphs with a unique 1-factor on $m\le 6$ vertices and determine their positive and negative invertibility