Inverses of trees (Q1066161)

Let \(G\) be a bipartite graph with non-singular adjacency matrix \(A\). For applications in Chemistry the least positive eigenvalue of \(A\) is of interest. The author proves a conjecture by I. Gutman that amongst all trees on \(2m\) vertices having perfect matchings, the path has smallest least positive eigenvalue. If \(G\) is a tree with a perfect matching it is shown that \(A^{-1}\) is diagonally similar to a \((0,1)\)-matrix. This result is used to obtain sharp bounds on the magnitude of the eigenvalue in question. A connection between the considered inversion problem and Möbius inversion on partially ordered sets is noted. The final section contains some open questions.