  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.
