Contributed talks on H- and M-MatricesThu 14:30–15:00, Room AV 91.21, Chair: Dawid Janse van Rensburg
In this communication we are concerned with the problem of characterizing when the group inverse of a singular, symmetric and irreducible $M$–matrix is also an $M$–matrix. This question has generated a great amount of works and has only been solved in a few specific cases, mainly involving the combinatorial laplacian of a network. Specifically, Y. Chen et al. established in  that the group inverse of a d.d. Jacobi matrix; that is the Laplacian associated with a weighted path, is also an $M$–matrix when its order is lower that $4$ and moreover under severe constraints on the weights. More generally, S.J. Kirkland et al. proved in  that the only weighted trees whose Laplacian matrix satisfy the required property are, in addition to the above paths, stars with some constraints on their conductances. Under this constraints they proved that there exist a infinity family of stars with the referred property. The use of Potential Theory techniques, has allowed to consider the problem with generality, avoiding the diagonal dominance hypothesis. Since each singular and symmetric and irreducible $M$–matrix can be identified with a singular and positive semidefinite Schrödinger operator on the connected graph associated with the matrix, the proposed problem consists on determine when the matrix associated with the corresponding Green operator is also an $M$–matrix. With this point of view, we obtained a necessary and sufficient condition for the general case, that allowed us to show that there exist infinitely many singular, symmetric and irreducible Jacobi matrices of any order such that its group inverse is also an $M$–matrix, see .
The aim of this work is to analyze the question for singular semidefinite Schrödinger operators on stars, or more generally on trees, and cycles.
This research was supported by the Spanish Research Council under project MTM2014-60450-R.
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