20th Conference of the International Linear Algebra Society
Date of publication
2016
Presentation's date
2016-07-15
Book of congress proceedings
Proceedings of the 20th ILAS Conference, 11-15 july 2016, Leuven
First page
158
Last page
158
Abstract
We study a particular case of self-adjoint operators on finite connected networks. Schrödinger operators define automorphisms on the subspace orthogonal to the eigenfunctions associated with the lowest eigenvalue. The inverse of an Schrödinger operator is known as orthogonal Green operator and it is closely related with some important indexes, as the Kirchhoff index or the effective resistances of the network. In preliminar works, several effects on these indexes of the original network have b...
We study a particular case of self-adjoint operators on finite connected networks. Schrödinger operators define automorphisms on the subspace orthogonal to the eigenfunctions associated with the lowest eigenvalue. The inverse of an Schrödinger operator is known as orthogonal Green operator and it is closely related with some important indexes, as the Kirchhoff index or the effective resistances of the network. In preliminar works, several effects on these indexes of the original network have been studied: deletion and contraction of vertices (see [3]), perturbations of edges (see [1]), addition of new vertices (see [2]), etc. In this work, we relate the Schrödinger operator of the network obtained by connecting two networks, in terms of the Schrödinger operators on the two initial networks. Moreover, the Green function of the new network can be computed in terms of the Green functions of the original networks. As a consequence, the Kirchhoff index of the new network can be expressed in terms of the Kirchhoff indexes of the two initial networks.