scholarly journals Further Results on Resistance Distance and Kirchhoff Index in Electric Networks

2016 ◽  
Vol 2016 ◽  
pp. 1-9
Author(s):  
Qun Liu ◽  
Jia-Bao Liu ◽  
Jinde Cao
Keyword(s):  
Electric Circuit ◽  
Circuit Theory ◽  
Effective Resistance ◽  
Electric Networks ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Disjoint

In electric circuit theory, it is of great interest to compute the effective resistance between any pairs of vertices of a network, as well as the Kirchhoff index. LetQ(G)be the graph obtained fromGby inserting a new vertex into every edge ofGand by joining by edges those pairs of these new vertices which lie on adjacent edges ofG. The set of such new vertices is denoted byI(G). TheQ-vertex corona ofG1andG2, denoted byG1⊙QG2, is the graph obtained from vertex disjointQ(G1)andVG1copies ofG2by joining theith vertex ofV(G1)to every vertex in theith copy ofG2. TheQ-edge corona ofG1andG2, denoted byG1⊖QG2, is the graph obtained from vertex disjointQ(G1)andIG1copies ofG2by joining theith vertex ofI(G1)to every vertex in theith copy ofG2. The objective of the present work is to obtain the resistance distance and Kirchhoff index for composite networks such asQ-vertex corona andQ-edge corona networks.

Resistance Distances in Vertex-Face Graphs

2018 ◽  
Vol 73 (2) ◽  
pp. 105-112 ◽  
Author(s):  
Yingmin Shangguan ◽  
Haiyan Chen
Keyword(s):  
Graph Theory ◽  
Sum Rules ◽  
Classical Problem ◽  
Electric Circuit ◽  
Circuit Theory ◽  
Orientable Surface ◽  
Resistance Distance

AbstractThe computation of two-point resistances in networks is a classical problem in electric circuit theory and graph theory. Let G be a triangulation graph with n vertices embedded on an orientable surface. Define K(G) to be the graph obtained from G by inserting a new vertex vϕ to each face ϕ of G and adding three new edges (u, vϕ), (v, vϕ) and (w, vϕ), where u, v and w are three vertices on the boundary of ϕ. In this paper, using star-triangle transformation and resistance local-sum rules, explicit relations between resistance distances in K(G) and those in G are obtained. These relations enable us to compute resistance distance between any two points of Kk(G) recursively. As explanation examples, some resistances in several networks are computed, including the modified Apollonian network and networks constructed from tetrahedron, octahedron and icosahedron, respectively.

scholarly journals The Extremal Cacti on Multiplicative Degree-Kirchhoff Index

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 83
Author(s):  
Fangguo He ◽  
Zhongxun Zhu
Keyword(s):  
Effective Resistance ◽  
Extremal Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set

For a graph G, the resistance distance r G ( x , y ) is defined to be the effective resistance between vertices x and y, the multiplicative degree-Kirchhoff index R ∗ ( G ) = ∑ { x , y } ⊂ V ( G ) d G ( x ) d G ( y ) r G ( x , y ) , where d G ( x ) is the degree of vertex x, and V ( G ) denotes the vertex set of G. L. Feng et al. obtained the element in C a c t ( n ; t ) with first-minimum multiplicative degree-Kirchhoff index. In this paper, we first give some transformations on R ∗ ( G ) , and then, by these transformations, the second-minimum multiplicative degree-Kirchhoff index and the corresponding extremal graph are determined, respectively.

Resistance Distances and Kirchhoff Index in Generalised Join Graphs

2017 ◽  
Vol 72 (3) ◽  
pp. 207-215 ◽  
Author(s):  
Haiyan Chen
Keyword(s):  
Complete Graph ◽  
Connected Graph ◽  
Effective Resistance ◽  
Electrical Network ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Join Graph ◽  
Explicit Formulae

AbstractThe resistance distance between any two vertices of a connected graph is defined as the effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. The Kirchhoff index of a graph is defined as the sum of all the resistance distances between any pair of vertices of the graph. Let G=H[G1, G2, …, Gk ] be the generalised join graph of G1, G2, …, Gk determined by H. In this paper, we first give formulae for resistance distances and Kirchhoff index of G in terms of parameters of ${G'_i}s$ and H. Then, we show that computing resistance distances and Kirchhoff index of G can be decomposed into simpler ones. Finally, we obtain explicit formulae for resistance distances and Kirchhoff index of G when ${G'_i}s$ and H take some special graphs, such as the complete graph, the path, and the cycle.

Kirchhoff Index of Cyclopolyacenes

2010 ◽  
Vol 65 (10) ◽  
pp. 865-870 ◽  
Author(s):  
Yan Wang ◽  
Wenwen Zhang
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Effective Resistance ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Hexagonal Chain

The resistance distance between two vertices of a connected graph G is computed as the effective resistance between them in the corresponding network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices. In this paper, following the method of Y. J. Yang and H. P. Zhang in the proof of the Kirchhoff index of the linear hexagonal chain, we obtain the Kirchhoff index of cyclopolyacenes, denoted by HRn, in terms of its Laplacian spectrum. We show that the Kirchhoff index of HRnis approximately one third of its Wiener index.

A Note on the Kirchhoff and Additive Degree-Kirchhoff Indices of Graphs

2015 ◽  
Vol 70 (6) ◽  
pp. 459-463 ◽  
Author(s):  
Yujun Yang ◽  
Douglas J. Klein
Keyword(s):  
Upper Bound ◽  
Extremal Graphs ◽  
Graph Invariants ◽  
Kirchhoff Index ◽  
Resistance Distance

AbstractTwo resistance-distance-based graph invariants, namely, the Kirchhoff index and the additive degree-Kirchhoff index, are studied. A relation between them is established, with inequalities for the additive degree-Kirchhoff index arising via the Kirchhoff index along with minimum, maximum, and average degrees. Bounds for the Kirchhoff and additive degree-Kirchhoff indices are also determined, and extremal graphs are characterised. In addition, an upper bound for the additive degree-Kirchhoff index is established to improve a previously known result.

scholarly journals Resistance Distance and Kirchhoff Index of Graphs with Pockets

2018 ◽  
Author(s):  
Qun Liu ◽  
Jiabao Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

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.

Calibration of Hydra Impedance Probes using Electric Circuit Theory

2009 ◽  
Vol 73 (2) ◽  
pp. 453-465 ◽  
Author(s):  
T. J. Kelleners ◽  
E. S. Ferre-Pikal ◽  
M. G. Schaap ◽  
G. B. Paige
Keyword(s):  
Electric Circuit ◽  
Circuit Theory
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