Preprint Article Version 1 This version is not peer-reviewed

Resistance Distance and Kirchhoff Index of Graphs with Pockets

Version 1 : Received: 9 October 2018 / Approved: 11 October 2018 / Online: 11 October 2018 (14:04:39 CEST)

How to cite: Liu, Q.; Liu, J. Resistance Distance and Kirchhoff Index of Graphs with Pockets. Preprints 2018, 2018100241 (doi: 10.20944/preprints201810.0241.v1). Liu, Q.; Liu, J. Resistance Distance and Kirchhoff Index of Graphs with Pockets. Preprints 2018, 2018100241 (doi: 10.20944/preprints201810.0241.v1).

Abstract

Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n ≥ 1,Vk= {v1,v2,··· ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m ≥ 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n ≥ 2, Ek= {e1,e2,···ek} is a subset of the edge set of F and Huvis a simple graph of order m ≥ 3, uv is a specified edge of Huvsuch that Huv− u is isomorphic to Huv− v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

Subject Areas

Kirchhoff index; resistance distance; generalized inverse.

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