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 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.

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 Infinite weighted graphs with bounded resistance metric

2018 ◽  
Vol 123 (1) ◽  
pp. 5-38
Author(s):  
Palle Jorgensen ◽  
Feng Tian
Keyword(s):  
Harmonic Functions ◽  
Voltage Drop ◽  
Interpolation Formula ◽  
Finite Energy ◽  
Boundary Representation ◽  
Special Importance ◽  
Weighted Graphs ◽  
Resistance Distance ◽  
Lipschitz Continuous ◽  
Vertex Set

We consider infinite weighted graphs $G$, i.e., sets of vertices $V$, and edges $E$ assumed countably infinite. An assignment of weights is a positive symmetric function $c$ on $E$ (the edge-set), conductance. From this, one naturally defines a reversible Markov process, and a corresponding Laplace operator acting on functions on $V$, voltage distributions. The harmonic functions are of special importance. We establish explicit boundary representations for the harmonic functions on $G$ of finite energy.We compute a resistance metric $d$ from a given conductance function. (The resistance distance $d(x,y)$ between two vertices $x$ and $y$ is the voltage drop from $x$ to $y$, which is induced by the given assignment of resistors when $1$ amp is inserted at the vertex $x$, and then extracted again at $y$.)We study the class of models where this resistance metric is bounded. We show that then the finite-energy functions form an algebra of ${1}/{2}$-Lipschitz-continuous and bounded functions on $V$, relative to the metric $d$. We further show that, in this case, the metric completion $M$ of $(V,d)$ is automatically compact, and that the vertex-set $V$ is open in $M$. We obtain a Poisson boundary-representation for the harmonic functions of finite energy, and an interpolation formula for every function on $V$ of finite energy. We further compare $M$ to other compactifications; e.g., to certain path-space models.

scholarly journals Resistance distance in directed cactus graphs

2020 ◽  
Vol 36 (36) ◽  
pp. 277-292
Author(s):  
R. Balaji ◽  
R.B. Bapat ◽  
Shivani Goel
Keyword(s):  
Laplacian Matrix ◽  
Minimum Length ◽  
Resistance Distance ◽  
Numerical Examples ◽  
Directed Paths ◽  
Strongly Connected ◽  
Vertex Set ◽  
Cactus Graphs ◽  
Classical Distance

Let $G=(V,E)$ be a strongly connected and balanced digraph with vertex set $V=\{1,\dotsc,n\}$. The classical distance $d_{ij}$ between any two vertices $i$ and $j$ in $G$ is the minimum length of all the directed paths joining $i$ and $j$. The resistance distance (or, simply the resistance) between any two vertices $i$ and $j$ in $V$ is defined by $r_{ij}:=l_{ii}^{\dagger}+l_{jj}^{\dagger}-2l_{ij}^{\dagger}$, where $l_{pq}^{\dagger}$ is the $(p,q)^{\rm th}$ entry of the Moore-Penrose inverse of $L$ which is the Laplacian matrix of $G$. In practice, the resistance $r_{ij}$ is more significant than the classical distance. One reason for this is, numerical examples show that the resistance distance between $i$ and $j$ is always less than or equal to the classical distance, i.e., $r_{ij} \leq d_{ij}$. However, no proof for this inequality is known. In this paper, it is shown that this inequality holds for all directed cactus graphs.

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 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.

A Variable Weight Function Model in Geological Variables Function Research

2013 ◽  
Vol 785-786 ◽  
pp. 1423-1429
Author(s):  
Wen Bo Liu ◽  
Lai Jun Liu
Keyword(s):  
Weight Function ◽  
Mineral Resources ◽  
Process Variable ◽  
Mathematics Model ◽  
Constant Weight ◽  
Connection Degree ◽  
State Variable ◽  
Variable Weight ◽  
The Status ◽  
Constant Coefficients

In mineral resources prediction and other research of geological variables, stability exactness of quantitative models concern modeling conditions, geological variables from model and the status of the variable. In traditional geological modeling process, variable support is measured under some contrains weight and this kind of weight is characterized by constant coefficients. Constant weight[1] has some limitations due to structuredness and dependency of variable. For overcoming the inflexibility of constant weight, this paper proposes geological variable mathematics model basedd state variable vector. We revise existing form of state variable weight and provide logarithm state variable vector as measurement level of geological variable weight coefficients. According to 1:200000 scale geochemistry measured data from Baishan area, we calculate the samples unit connection degree based on exponent and logarithm state variable vector and compare the connection degree based on constant weight. The connection degree sorting has the similarity as a whole among them, but there is the obvious difference locally. We can conclude that geological variable weight function based on state variable vector is more flexible and fine.

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 2-Approximating Feedback Vertex Set in Tournaments

2021 ◽  
Vol 17 (2) ◽  
pp. 1-14
Author(s):  
Daniel Lokshtanov ◽  
Pranabendu Misra ◽  
Joydeep Mukherjee ◽  
Fahad Panolan ◽  
Geevarghese Philip ◽  
...  
Keyword(s):  
Approximation Algorithm ◽  
Weight Function ◽  
Directed Graph ◽  
Polynomial Time ◽  
Optimal Solutions ◽  
Feedback Vertex Set ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
Unique Games Conjecture ◽  
Unique Games

A tournament is a directed graph T such that every pair of vertices is connected by an arc. A feedback vertex set is a set S of vertices in T such that T − S is acyclic. We consider the Feedback Vertex Set problem in tournaments. Here, the input is a tournament T and a weight function w : V ( T ) → N, and the task is to find a feedback vertex set S in T minimizing w ( S ) = ∑ v∈S w ( v ). Rounding optimal solutions to the natural LP-relaxation of this problem yields a simple 3-approximation algorithm. This has been improved to 2.5 by Cai et al. [SICOMP 2000], and subsequently to 7/3 by Mnich et al. [ESA 2016]. In this article, we give the first polynomial time factor 2-approximation algorithm for this problem. Assuming the Unique Games Conjecture, this is the best possible approximation ratio achievable in polynomial time.

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