Linear algebra and its applications

Vol. 576, p. 5-34

DOI: 10.1016/j.laa.2018.01.026

Date of publication: 2019-09-01

Abstract:

In this paper we introduce new effective resistances on a given network, associated with a positive value and two weights on the vertex set and investigate under which conditions they determine a distance. We prove that this property is closely related with superharmonicity. Moreover, we analyze the behavior of these distances under the usual network transformations, specially the so-called star-mesh transformation. We also compute the effective resistance for an extended network; that is, the network obtained from the former one by joining a new vertex, and then study the effect of the contraction of this new network; that is, we apply a star-mesh transformation with center in the joined vertex.]]>

Linear algebra and its applications

Vol. 491, p. 419-433

DOI: 10.1016/j.laa.2015.11.012

Date of publication: 2016-02-15

Abstract:

Any elliptic operator defines an automorphism on the orthogonal subspace to the eigenfunctions associated with the lowest eigenvalue, whose inverse is the orthogonal Green operator. In this study, we show that elliptic Schrödinger operators on networks that have been obtained by adding a new vertex to a given network, can be seen as perturbations of the Schrödinger operators on the initial network. Therefore, the Green function of the new network can be computed in terms of the Green function of the original network.]]>

Linear algebra and its applications

Vol. 468, p. 270-285

DOI: 10.1016/j.laa.2014.10.042

Date of publication: 2015

Linear algebra and its applications

Vol. 468, p. 38-47

DOI: 10.1016/j.laa.2013.12.039

Date of publication: 2015

Abstract:

A polyomino is an edge-connected union of cells in the planar square lattice. Here we consider generalized linear polyominoes; that is, the polyominoes supported by an n Ã— 2 lattice. In this paper, we obtain the Green function and the Kirchhoff index of a generalized linear polyomino as a perturbation of a 2n-path by adding weighted edges between opposite vertices. This approach deeply links generalized linear polyomino Green functions with the inverse M-matrix problem, and especially with the so-called Green matrices.

A polyomino is an edge-connected union of cells in the planar square lattice. Here we consider generalized linear polyominoes; that is, the polyominoes supported by an n Ã— 2 lattice. In this paper, we obtain the Green function and the Kirchhoff index of a generalized linear polyomino as a perturbation of a 2n-path by adding weighted edges between opposite vertices. This approach deeply links generalized linear polyomino Green functions with the inverse M-matrix problem, and especially with the so-called Green matrices.]]>

Linear algebra and its applications

Vol. 442, p. 115-134

Date of publication: 2014

Linear algebra and its applications

Vol. 436, num. 5, p. 1090-1098

DOI: 10.1016/j.laa.2011.06.044

Date of publication: 2012-03-01

Linear algebra and its applications

Vol. 432, num. 9, p. 2278-2292

DOI: 10.1016/j.laa.2009.05.032

Date of publication: 2010-04-15

Abstract:

In this work we define the effective resistance between any pair of vertices with respect to a value λ ≥ 0 and a weight ω on the vertex set. This allows us to consider a generalization of the Kirchhoff Index of a finite network. It turns out that λ is the lowest eigenvalue of a suitable semi-definite positive Schrödinger operator and ω is the associated eigenfunction. We obtain the relation between the effective resistance, and hence between the Kirchhoff Index, with respect to λ and ω and the eigenvalues of the associated Schrödinger operator. However, our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network. From these expressions, we derive a full generalization of Foster’s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk. Finally, we compute the effective resistances and the generalized Kirchhoff Index with respect to a λ and ω for some families of networks with symmetries, specifically for weighted wagon-wheels and circular ladders.]]>

Linear algebra and its applications

Vol. 432, num. 9, p. 2438-2454

DOI: 10.1016/j.laa.2009.11.008

Date of publication: 2010-04-15

Linear algebra and its applications

Vol. 429, num. 7, p. 1823-1839

Date of publication: 2008-10

Linear algebra and its applications

Vol. 423, num. 1, p. 109-118

DOI: 10.1016/j.laa.2006.11.018

Date of publication: 2007-05