Inverses of trees

LetT be a tree with a perfect matching. It is known that in this case the adjacency matrixA ofT is invertible and thatA ^?1 is a (0, 1, ?1)-matrix. We show that in factA ^?1 is diagonally similar to a (0, 1)-matrix, hence to the adjacency matrix of a graph. We use this to provide sharp bounds on the least positive eigenvalue ofA and some general information concerning the behaviour of this eigenvalue. Some open problems raised by this work and connections with Möbius inversion on partially ordered sets are also discussed.