Overdetermined partial resolvent kernels for generalized cylinders
Author
Carmona, A.; Encinas, A.; Mitjana, M.
Type of activity
Presentation of work at congresses
Name of edition
Spectra of Graphs and Applications 2016
Date of publication
2016
Presentation's date
2016-05-19
Book of congress proceedings
Spectra of graphs and applications 2016 (SGA): Belgrade, Serbia: May 18–20, 2016: book of abstracts
First page
22
Last page
23
Abstract
The resistive distance has become an useful tool to analyze struc- tural properties of networks. Unlike the geodesic distance , the resis- tive distance takes into account all paths between vertices. The high sensibility of this metric with respect to small perturbations, makes it suitable to compare different network structures. This is one of the main reason for which effective resistances and the correspond- ing Kirchhoff Index, have emerged as structure-descriptors in Organic Chemistry, wher...
The resistive distance has become an useful tool to analyze struc- tural properties of networks. Unlike the geodesic distance , the resis- tive distance takes into account all paths between vertices. The high sensibility of this metric with respect to small perturbations, makes it suitable to compare different network structures. This is one of the main reason for which effective resistances and the correspond- ing Kirchhoff Index, have emerged as structure-descriptors in Organic Chemistry, where the topology of chemical compounds is represented by a molecular network where edge weights correspond to bond prop- erties. Effective resistances can be expressed in terms of the group inverse of the Laplacian matrix and the corresponding Kirchhoff index is nothing else but its trace. Other distances on networks have been studied with similar pur- poses. In this work, we pay attention on the so–called forest metric introduced by P. Chebotarev and E. Shamis at late 90’s, or more specif- ically on its reformulation as adjusted forest metric , that they interpret as a measure of the accessibility . The adjusted forests metrics form a one–parametric family, where the parameter determines the proportion of taking into account long and short routes between vertices. More- over, the usual resistive distance is the asymptotic value of the metric when the parameter goes to infinite. In this communication we show that each distance of the one–parametric family is a resistive distance corresponding to an Schr ¨odinger operator on the network. We also study the effect of some perturbation on the operator on its resistive distance