Green's function in partial subdivision networks

ABSTRACT

In the present work, we define a partial subdivision network ${\mathrm{\Gamma }}^S$ of a given network Γ, by inserting a new vertex in some selected edges of Γ, so that each of these edges is replaced by two new edges with conductances that fulfil the Kirchhoff series law on the new network. Then, we obtain an expression for the Green kernel of the partial subdivision network in terms of the Green kernel of the base network. For that, we show the relation between Poisson problems on the partial subdivision network and Poisson problems on the base network. Moreover, we also obtain the effective resistance and the Kirchhoff index of the partial 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 star network in which we have subdivided the even edges.

KEYWORDS:

• Resistance distance
• Green kernel
• Kirchhoff index
• partial subdivision network

MSC SUBJECT CLASSIFICATIONS:

• 31C20
• 15A09
• 34B45

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Ángeles Carmona http://orcid.org/0000-0001-7713-1066

Margarida Mitjana http://orcid.org/0000-0002-6563-5512

Enric Monsó http://orcid.org/0000-0003-4815-6269

Additional information

Funding

This work has been partly supported by the Spanish Research Council (Comisión Interministerial de Ciencia y Tecnología) under project MTM2017-85996-R.