Some Two-Vertex Resistances of Nested Triangle Network
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Some Two-Vertex Resistances of Nested Triangle Network Muhammad Shoaib Sardar1
· Xiang-Feng Pan1
· Si-Ao Xu1
Received: 2 November 2019 / Revised: 28 August 2020 / Accepted: 4 September 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 so as to correspond to G, with each edge being replaced by a unit resistor. The Kirchhoff index of G is the sum of the resistance distances between all pairs of vertices of G. A planar graph is a nested triangle graph with 3n vertices constructed from a sequence of n triangles by joining the pairs of corresponding vertices on consecutive triangles in the sequence. In this paper, some two-vertex resistances of nested triangle network was procured by utilizing techniques from the theory of electrical networks, i.e., the series and parallel principles, the principle of substitution, the star-triangle transformation and the delta-wye transformation. And the Kirchhoff index of nested triangle network is given by the Laplacian eigenvalues. Keywords Resistance distance · Kirchhoff index · Network · Nested triangle network Mathematics Subject Classification 05C12
1 Introduction We recall some background and notations in Sect. 1.1. Section 2 contains some known results that lead to main results in Sect. 3. The conclusive remarks for whole paper are discussed in Sect. 4. For undetermined notations and terminologies, see the book by Bollobas ´ [1].
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Xiang-Feng Pan [email protected] Muhammad Shoaib Sardar [email protected] Si-Ao Xu [email protected]
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School of Mathematical Sciences, Anhui University, Hefei, Anhui 230601, People’s Republic of China
Circuits, Systems, and Signal Processing
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Fig. 1 The situation where the classical distance is indistinguishable
1.1 Background Let G = (V , E) be a connected graph with vertex-set V and edge-set E. The distance between two vertices u, v ∈ V is the length of shortest path between u and v. It is represented by d(u, v) and is largely studied. But this notion of distance is not always convenient. For example, d(u, v) = 2 in both graphs (see Fig. 1). But it is clearly seen that in Fig. 1b, there are more links between vertices u and v as compared to Fig. 1a. Therefore, it is logical to say that in some sense the distance between u and v in the graph of Fig. 1b is less than that of Fig. 1a. This feature is not expressed by a classical distance. Therefore, another notion for distance was given by Klein and Randic´ [18] called “resistance distance” for better communication. For two vertices u and v in a connected graph G, the resistance distance, denoted by r G (u, v), is defined as the effective resistance between u and v by replacing each edge of G with a unit resistor (resistor with resistance 1 Ohm). We abbreviate r G (u, v) as r (u, v) if the graph G is clear from context. For example, the cycle graph C3 and corres
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