Some Two-Vertex Resistances of the Three-Towers Hanoi Graph Formed by a Fractal Graph
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Some Two-Vertex Resistances of the Three-Towers Hanoi Graph Formed by a Fractal Graph Muhammad Shoaib Sardar1
· Xiang-Feng Pan1
· Yun-Xiang Li1
Received: 22 November 2019 / Accepted: 13 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The resistance distance between two vertices of a simple connected graph G is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to G, with each edge being replaced by a unit resistor. In this study, we apply the principle of substitution to three-towers Hanoi graph to procure an equivalent network that enables us to only use the series and parallel principles and the delta-wye transformation to acquire the some two-vertex resistance distances of three-towers Hanoi graph. Keywords Resistance distance · Network · Three-towers Hanoi graph Mathematics Subject Classification 05C12 · 05C35 · 05C81
1 Introduction The study of electric networks was developed by Kirchhoff more than 170 years ago [1]. Apart from the central problem in electric circuit theory, it is not easy to procure some twovertex resistances in some complex networks. But, there are several techniques from electrical network theory for computing the resistance distance, i.e., series and parallel principles, deltawye transformation, star-triangle transformation, and star-mesh transformation. There are also some formulae such as combinatorial formula [2], algebraic formulae [3–9], probabilistic formulae [2,10], and so on. Resistance distance has been computed for some particular graphs, for example, circulant graphs [11], Cayley graphs over finite Abelian groups [12], wheels and fans [13], regular graphs [14,15], pseudo-distance-regular graphs [16], some fullerene graphs [17], Sierpinski gasket network [18], ring network [19] and so on [20–25]. The fractal theory has become an increasingly popular topic of both discussion and research in recent years. Fractals are infinitely complex patterns that are self-similar across different scales. They created by repeating a simple process over and over in an ongoing feedback loop. The most typical example of self-similar fractal is Sierpinski gasket [18],
Communicated by Eric A. Carlen.
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Xiang-Feng Pan [email protected] School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, People’s Republic of China
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which resembles the three-towers Hanoi graph. It is a specific case of Hanoi graph that has been well studied after the work of Scorer et al. [26]. The diameter for three-towers Hanoi graph is 2n+1 − 1. But it is still an open problem in mathematics about the diameter and construction of Hanoi graphs if we have more than three-towers. For more details about the Hanoi graph, we refer the readers to the following articles [27–36]. The idea of resistance distance was introduced by Klein and Randi´c [37]. The resistance distance between two vertices of a simple connected graph, G, is equal to the resistance between two equivalent points on an elect
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