Computing the Green function for a Dirichlet problem on spider networks

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Matrix tree theorem for Schrödinger operators on networks

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The group inverse of subdivision networks

In this paper, given a network and a subdivision of it, we show how the Group Inverse of the subdivision network can be related to the Group Inverse of initial given network. Our approach establishes a relationship between solutions of related Poisson problems on both networks and takes advantatge on the definition of the Group Inverse matrix.Peer ReviewedPostprint (author's final draft

Effective resistances and Kirchhoff index in subdivision networks

We define a subdivision network ¿S of a given network ¿; by inserting a new vertex in every edge, so that each edge is replaced by two new edges with conductances that fulfill electrical conditions on the new network. In this work, we firstly obtain an expression for the Green kernel of the subdivision network in terms of the Green kernel of the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the subdivision network in terms of the corresponding parameters on the base network. Finally, as an example, we carry out the computations in the case of a wheel.Peer ReviewedPostprint (author's final draft

The effective resistance of extended or contracted networks

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Discrete inverse problem on grids

In this work, we present an algorithm to the recovery of the conductance of a n –dimensional grid. The algorithm is based in the solution of some overdetermined partial boundary value problems defined on the grid; that is, boundary value problem where the boundary conditions are set only in a part of the boundary (partial), and moreover in a fix subset of the boundary we prescribe both the value of the function and of its normal derivative (overdetermined). Our goal is to recover the conductance of a n –dimensional grid network with boundary using only boundary measurements and global equilibrium conditions. This problem is known as inverse boundary value problem . In general, inverse problems are exponentially ill–posed, since they are highly sensitive to changes in the boundary data. However, in this work we deal with a situation where the recovery of the conductance is feasible: grid networks. The recovery of the conductances of a grid network is performed here using its Schr ¨odinger matrix and boundary value problems associated to it. Moreover, we use the Dirichlet–to–Robin matrix, also known as response matrix of the network, which contains the boundary information. It is a certain Schur complement of the Schr ¨odinger matrix. The Schur complement plays an important role in matrix analysis, statistics, numerical analysis, and many other areas of mathematics and its applications.Postprint (author's final draft

Resistance distances on networks

This paper aims to study a family of distances in networks associated witheffective resistances. Speci cally, we consider the e ective resistance distance with respect to a positive parameter and a weight on the vertex set; that is, the effective resistance distance associated with an irreducible and symmetric M-matrix whose lowest eigenvalue is the parameter and the weight function is the associated eigenfunction. The main idea is to consider the network embedded in a host network with additional edges whose conductances are given in terms of the mentioned parameter. The novelty of these distances is that they take into account not only the influence of shortest and longest weighted paths but also the importance of the vertices. Finally, we prove that the adjusted forest metric introduced by P. Chebotarev and E. Shamis is nothing else but a distance associated with a Schr odinger operator with constant weightPeer ReviewedPostprint (author's final draft

Green’s kernel for subdivision networks

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Dirichlet-to-Robin matrix on networks

In this work, we de ne the Dirichlet{to{Robin matrix associated with a Schr odinger type matrix on general networks, and we prove that it satis es the alternating property which is essential to characterize those matrices that can be the response matrices of a network. We end with some examples of the sign pattern behavior of the alternating paths.Postprint (author's final draft

Green functions for boundary value problems on product networks

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