scholarly journals Resistance Distances in Linear Polyacene Graphs

2021 ◽  
Vol 8 ◽  
Author(s):  
Dayong Wang ◽  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Exact Expression ◽  
Effective Resistance ◽  
Electrical Network ◽  
Combinatorial Approach ◽  
Network Approach ◽  
Resistance Distance ◽  
Electric Network

The resistance distance between any two vertices of a connected graph is defined as the net effective resistance between them in the electrical network constructed from the graph by replacing each edge with a unit resistor. In this article, using electric network approach and combinatorial approach, we derive exact expression for resistance distances between any two vertices of polyacene graphs.

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 Some Bounds for the Kirchhoff Index of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Independence Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Electrical Network ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Fullerene Graphs

The resistance distance between two vertices of a connected graphGis defined as the effective resistance between them in the corresponding electrical network constructed fromGby replacing each edge ofGwith a unit resistor. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.

scholarly journals The first two cacti with larger multiplicative eccentricity resistance-distance

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1783-1800
Author(s):  
Yunchao Hong ◽  
Zhongxun Zhu
Keyword(s):  
Connected Graph ◽  
Effective Resistance ◽  
Extremal Graphs ◽  
Resistance Distance ◽  
G Value

For a connected graph G, the multiplicative eccentricity resistance-distance ?*R(G) is defined as ?*R(G) = ?{x,y}?V(G)?(x)??(y)RG(x,y), where ?(?) is the eccentricity of the corresponding vertex and RG(x,y) is the effective resistance between vertices x and y. A cactus is a connected graph in which any two simple cycles have at most one vertex in common. Let Cat(n;t) be the set of cacti possessing n vertices and t cycles, where 0 ? t ? n-1/2. In this paper, we first introduce some edge-grafting transformations which will increase ?*R(G). As their applications, the extremal graphs with maximum and second-maximum ?*R(G)-value in Cat(n,t) are characterized, respectively.

scholarly journals Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Wenjie Ning ◽  
Kun Wang ◽  
Hassan Raza
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Effective Resistance ◽  
Resistance Distance ◽  
Bicyclic Graph ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

Let G = V , E be a connected graph. The resistance distance between two vertices u and v in G , denoted by R G u , v , is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as D R G = ∑ u , v ⊆ V G d G u + d G v R G u , v , where d G u is the degree of a vertex u in G and R G u , v is the resistance distance between u and v in G . A bicyclic graph is a connected graph G = V , E with E = V + 1 . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n ≥ 6 vertices.

A Simple Method for Computing Resistance Distance

2003 ◽  
Vol 58 (9-10) ◽  
pp. 494-498 ◽  
Author(s):  
Ravindra B. Bapat ◽  
Ivan Gutmana ◽  
Wenjun Xiao
Keyword(s):  
Molecular Graph ◽  
Laplacian Matrix ◽  
Effective Resistance ◽  
Electrical Network ◽  
Resistance Distance ◽  
Simple Method

The resistance distance ri j between two vertices vi and vj of a (connected, molecular) graph G is equal to the effective resistance between the respective two points of an electrical network, constructed so as to correspond to G, such that the resistance of any edge is unity. We show how rij can be computed from the Laplacian matrix L of the graph G: Let L(i) and L(i, j) be obtained from L by deleting its i-th row and column, and by deleting its i-th and j-th rows and columns, respectively. Then rij = detL(i, j)/detL(i).

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.

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.

Dynast – Computer Program Used in Power Engineering

2013 ◽  
Vol 284-287 ◽  
pp. 1035-1038 ◽  
Author(s):  
Lucie Nohacova ◽  
Karel Nohac
Keyword(s):  
Computer Program ◽  
Power Engineering ◽  
Electrical Network ◽  
Mechanical Equipment ◽  
Modeling Tools ◽  
Electric Network ◽  
Simulation Tools

There are relatively insufficient quality specialized modeling tools for power engineering. Regardless there are few possibilities how to simulate main elements of electric network and systems consisting of them. Beside most used specialized tool EMTP-ATP there is MATLAB with its Simulink library SimPowerSystems and newly also DYNAST with its own growing library of power engineering elements. There are also many other general simulation tools with libraries of elements suitable for power engineering. Therefore there seems to be opportunity to compare quality of mentioned tools in accuracy, reliability, speed and flexibility in high demanding problems. Branch “Power engineering” is very specific, because there is necessary in most cases to simulate multiple technical disciplines. Not only electrical network is to solve, but also connection with mechanical equipment and other devices of power engineering. One of the most powerful tools that meet all these requirements is Dynast. Now new library for electric components was developed to make creation of model easy and fast.

Potential Theory on Distance-Regular Graphs

1993 ◽  
Vol 2 (3) ◽  
pp. 243-255 ◽  
Author(s):  
Norman L. Biggs
Keyword(s):  
Random Walks ◽  
Potential Theory ◽  
Effective Resistance ◽  
Electrical Network ◽  
Regular Graphs ◽  
Regular Case ◽  
Explicit Formulae ◽  
Distance Regular Graphs ◽  
A Current

A graph may be regarded as an electrical network in which each edge has unit resistance. We obtain explicit formulae for the effective resistance of the network when a current enters at one vertex and leaves at another in the distance-regular case. A well-known link with random walks motivates a conjecture about the maximum effective resistance. Arguments are given that point to the truth of the conjecture for all known distance-regular graphs.

scholarly journals On the Resistance Matrix of a Graph

10.37236/5295 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Jiang Zhou ◽  
Zhongyu Wang ◽  
Changjiang Bu
Keyword(s):  
Regular Graph ◽  
Connected Graph ◽  
Spanning Trees ◽  
Regular Graphs ◽  
Resistance Distance ◽  
Strongly Regular ◽  
Resistance Matrix ◽  
Number Of Spanning Trees

Let $G$ be a connected graph of order $n$. The resistance matrix of $G$ is defined as $R_G=(r_{ij}(G))_{n\times n}$, where $r_{ij}(G)$ is the resistance distance between two vertices $i$ and $j$ in $G$. Eigenvalues of $R_G$ are called R-eigenvalues of $G$. If all row sums of $R_G$ are equal, then $G$ is called resistance-regular. For any connected graph $G$, we show that $R_G$ determines the structure of $G$ up to isomorphism. Moreover, the structure of $G$ or the number of spanning trees of $G$ is determined by partial entries of $R_G$ under certain conditions. We give some characterizations of resistance-regular graphs and graphs with few distinct R-eigenvalues. For a connected regular graph $G$ with diameter at least $2$, we show that $G$ is strongly regular if and only if there exist $c_1,c_2$ such that $r_{ij}(G)=c_1$ for any adjacent vertices $i,j\in V(G)$, and $r_{ij}(G)=c_2$ for any non-adjacent vertices $i,j\in V(G)$.

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