Some results on Electrical networks in graph theory

Proceedings of the IEEE, 2018

Algebraic graph theory is a cornerstone in the study of electrical networks ranging from miniature integrated circuits to continental-scale power systems. Conversely, many fundamental results of algebraic graph theory were laid out by early electrical circuit analysts. In this paper we survey some fundamental and historic as well as recent results on how algebraic graph theory informs electrical network analysis, dynamics, and design. In particular, we review the algebraic and spectral properties of graph adjacency, Laplacian, incidence, and resistance matrices and how they relate to the analysis, networkreduction, and dynamics of certain classes of electrical networks. We study these relations for models of increasing complexity ranging from static resistive DC circuits, over dynamic RLC circuits, to nonlinear AC power flow. We conclude this paper by presenting a set of fundamental open questions at the intersection of algebraic graph theory and electrical networks.

Electrical Network Topology, Electrical Network Graph Theory, Node, Branch, Twig, Link, Tree, Cotree

IOSR Journals , 2019

In this paper we present a circuit network in the concept of application of graph theory and circuit models of graph are represented in logical connection by using truth table. We formulate the matrix method of adjacency and incidence of matrix followed by application of truth table.

Springer eBooks, 1972

Electrical Network Topology/Electrical Network Graph Theory: Complete Incidence Matrix, Reduced or Incidence Matrices of Electrical Network, Possible trees that can be drawn out of a given graph

Electrical Network Topology, Electrical Network Graph Theory, Incidence Matrices of Electrical Network [Complete incidence and reduced incidence matrices]

When we use linear algebra to understand physical systems, we often find more structure in the matrices and vectors than appears in the examples we make up in class. There are many applications of linear algebra; for example, chemists might use row reduction to get a clearer picture of what elements go into a complicated reaction. In this lecture we explore the linear algebra associated with electrical networks.

Discrete Mathematics, 1986

Voltage graphs, one of the main tools for constructing graph embeddings, appear to be useful in various areas of graph theory and combinatorics. The present paper concerns several problems in voltage graph theory such as equivalence and regularity of coverings generated by (permutation) voltage graphs, automorphism groups and some other things.

2004

A metrized graph is a finite weighted graph whose edges are thought of as line segments. In this expository paper, we study the Laplacian operator on a metrized graph and some important functions related to it, including the ``j-function'', the effective resistance, and eigenfunctions of the Laplacian. We discuss the relationship between metrized graphs and electrical networks, which provides some

Linear Algebra and its Applications, 1998

We consider circular planar graphs and circular planar resistor networks. Associated with each circular planar graph F there is a set n(F) = { (P; Q) } of pairs of sequences of boundary nodes which are connected through F. A graph F is called critical if removing any edge breaks at least one of the connections (P: Q) in n(F). We prove that two critical circular planar graphs are Y-A equivalent if and only if they have the same connections. If a conductivity ;, is assigned to each edge in F, there is a linear from boundary voltages to boundary currents, called the network response. This linear map is represented by a matrix A:. We show that if (F,7) is any circular planar resistor network whose underlying graph F is critical, then the values of all the conductors in F may be calculated from A. Finally, we give an algebraic description of the set ot" network response matrices that can occur for circular planar resistor networks.