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Effective resistances and Kirchhoff index in subdivision networks

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Cita com:
hdl:2117/107170

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Carmona Mejías, ÁngelesMés informacióMés informacióMés informació
Monsó Burgués, Enrique P.J.Més informacióMés informacióMés informació
Mitjana Riera, MargaridaMés informacióMés informacióMés informació
Document typeConference report
Defense date2016
Rights accessRestricted access - publisher's policy
All rights reserved. This work is protected by the corresponding intellectual and industrial property rights. Without prejudice to any existing legal exemptions, reproduction, distribution, public communication or transformation of this work are prohibited without permission of the copyright holder
Abstract
In this work we compute the effective resistances and the Kirchhoff Index of subdivision networks in terms of the corresponding parameters of the original network. Our techniques are based on the study of discrete operators using discrete Potential Theory. Starting from a given network G = ( V, E, c ), we add a new vertex v xy at every edge { x, y } ¿ E and define new conductances c ( x, v xy ) so as to satisfy the electrical compatibility condition 1 c ( x, y ) = 1 c ( x, v xy ) + 1 c ( y, v xy ) , in order to obtain a subdivision network G S = ( V S , E S , c ) . In this setting we prove how the solution to a compatible Poisson problem on the subdivision network G S can be related with the solution of a suitable Poisson problem on the inicial network G . We use contraction and extension of functions from C ( V S ) to C ( V ) and viceversa, respectively to achieve our result. As effective resistances are computed with the aid of a solution (no matter what) to a particular Poisson problem, we can stablish an affine relationship between effective resistances defined for vertices in G S and the effective resistances defined for vertices in G . Here the coefficients are simply weighted averages of the conductances previously stated. Finally we relate the Kirchhoff index of the subdivision network K (G S ) with the Kirchhoff index of the initial network K (G) by using the so called Green‘s kernel function because the desired parameter can be computed as the trace of the kernel.
CitationCarmona, A., Monso, E., Mitjana, M. Effective resistances and Kirchhoff index in subdivision networks. A: European Congress of Mathematics. "7ECM Berlin 2016: 7th European Congress of Mathematics, July 18-22, 2016, Technische Universität Berlin". Berlin: 2016, p. 70. 
URIhttp://hdl.handle.net/2117/107170
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  • Departament de Matemàtiques - Ponències/Comunicacions de congressos [1.057]
  • COMPTHE - Combinatòria i Teoria Discreta del Potencial pel control de paràmetres en xarxes - Ponències/Comunicacions de congressos [18]
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