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.

Degree Kirchhoff Index of Bicyclic Graphs

2017 ◽  
Vol 60 (1) ◽  
pp. 197-205 ◽  
Author(s):  
Zikai Tang ◽  
Hanyuan Deng
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Degree Of Vertex ◽  
Vertex Set ◽  
Maximum And Minimum Degree ◽  
Bicyclic Graphs

AbstractLet G be a connected graph with vertex set V(G).The degree Kirchhoò index of G is defined as S'(G) = Σ{u,v}⊆V(G) d(u)d(v)R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between vertices u and v. In this paper, we characterize the graphs having maximum and minimum degree Kirchhoò index among all n-vertex bicyclic graphs with exactly two cycles.

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.

scholarly journals Resistance distances on networks

2017 ◽  
Vol 11 (1) ◽  
pp. 136-147 ◽  
Author(s):  
A. Carmona ◽  
A.M. Encinas ◽  
M. Mitjana
Keyword(s):  
Weight Function ◽  
Main Idea ◽  
Effective Resistance ◽  
Positive Parameter ◽  
Resistance Distance ◽  
Constant Weight ◽  
Vertex Set ◽  
Weighted Paths ◽  
Lowest Eigenvalue

This paper aims to study a family of distances in networks associated with effective resistances. Specifically, we consider the effective resistance distance with respect to a positive parameter and a weight on the vertex set; that is, the effective resistance distance associated with an irreducible and symmetric M-matrix whose lowest eigenvalue is the parameter and the weight function is the associated eigenfunction. The main idea is to consider the network embedded in a host network with additional edges whose conductances are given in terms of the mentioned parameter. The novelty of these distances is that they take into account not only the influence of shortest and longest weighted paths but also the importance of the vertices. Finally, we prove that the adjusted forest metric introduced by P. Chebotarev and E. Shamis is nothing else but a distance associated with a Schr?dinger operator with constant weight.

scholarly journals Resistance Distance and Kirchhoff Index for a Class of Graphs

2018 ◽  
Vol 2018 ◽  
pp. 1-8
Author(s):  
WanJun Yin ◽  
ZhengFeng Ming ◽  
Qun Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

Let G[F,Vk,Hv] 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, Hv is a simple graph of order m≥2, and 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, Huv is a simple graph of order m≥3, and uv is a specified edge of Huv such 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.

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.

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.

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 Maximum Reciprocal Degree Resistance Distance Index of Bicyclic Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-14
Author(s):  
Gaixiang Cai ◽  
Xing-Xing Li ◽  
Guidong Yu
Keyword(s):  
Connected Graph ◽  
Extremal Graph ◽  
Resistance Distance ◽  
Bicyclic Graphs

The reciprocal degree resistance distance index of a connected graph G is defined as RDR G = ∑ u , v ⊆ V G d G u + d G v / r G u , v , where r G u , v is the resistance distance between vertices u and v in G . Let ℬ n denote the set of bicyclic graphs without common edges and with n vertices. We study the graph with the maximum reciprocal degree resistance distance index among all graphs in ℬ n and characterize the corresponding extremal graph.

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