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Circuits, Systems, and Signal Processing

16.05.2024

On the Resistance Distance and Kirchhoff Index of \(K_n\)-chain(Ring) Network

verfasst von: Wensheng Sun, Muhammad Shoaib Sardar, Yujun Yang, Shou-Jun Xu

Erschienen in: Circuits, Systems, and Signal Processing

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Abstract

The resistance distance \(r_{G}(u,v)\) between two vertices u and v of a graph G is defined as the net effective resistance between them in the electric network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index Kf(G) is defined as the sum of resistance distances between all pairs of vertices. Let \(L^{m}_{n}\) be a \(K_n\)-chain network with m complete graphs. Then identifying the opposite lateral edges of \(L^{m}_{n}\) in an order way yields the \(K_n\)-ring, denoted by \(C^{m}_{n}\). In this paper, we first construct a new equivalent network transformation on complete graphs. Then utilize combinatorial and electrical network approaches, we give explicit formula for the resistance distances between any two vertices in \(L^{m}_{n}\) and \(C^{m}_{n}\). Further, the closed-form formulas of the Kirchhoff index for \(L^{m}_{n}\) and \(C^{m}_{n}\) are also obtained. In addition, our results contain the main results of [Symmetry. 15(5) (2023) 1122] and [Phys. Scr. 98(4) (2023) 045222] as special cases.

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Metadaten
Titel
On the Resistance Distance and Kirchhoff Index of -chain(Ring) Network
verfasst von
Wensheng Sun
Muhammad Shoaib Sardar
Yujun Yang
Shou-Jun Xu
Publikationsdatum
16.05.2024
Verlag
Springer US
Erschienen in
Circuits, Systems, and Signal Processing
Print ISSN: 0278-081X
Elektronische ISSN: 1531-5878
DOI
https://doi.org/10.1007/s00034-024-02709-y