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Godsil (1985) defined a graph to be invertible if it has a non-singularadjacency matrix whose inverse is diagonally similar to a nonnegative integralmatrix; the graph defined by the last matrix is then the inverse of theoriginal graph. In this paper we call such graphs positively invertible andintroduce a new concept of a negatively invertible graph by replacing theadjective `nonnegative' by `nonpositive in Godsil's definition; the graphdefined by the negative of the resulting matrix is then the negative inverse ofthe original graph. We propose new constructions of integrally invertiblegraphs (those with non-singular adjacency matrix whose inverse is integral)based on an operation of `bridging' a pair of integrally invertible graphs oversubsets of their vertices, with sufficient conditions for their positive andnegative invertibility. We also analyze spectral properties of graphs arisingfrom bridging and derive lower bounds for their least positive eigenvalue. Asan illustration we present a census of graphs with a unique 1-factor on $m\le6$ vertices and determine their positive and negative invertibility.

On a Construction of Integrally Invertible Graphs and their Spectral Properties