Cycles, Bonds, and Electrical Networks

Part of the Undergraduate Texts in Mathematics book series (UTM)


In this chapter we will deal with some interesting linear algebra related to the structure of a directed graph. Let D=(V,E) be a digraph. A function \(f : E \rightarrow \mathbb{R}\) is called a circulation if for every vertex vV we have
$$\displaystyle{ \sum _{{ e\in E \atop \mathrm{init}(e)=v} }f(e) =\sum _{{ e\in E \atop \mathrm{fin}(e)=v} }f(e). }$$
Thus if we think of the edges as pipes and f as measuring the flow (quantity per unit of time) of some commodity (such as oil) through the pipes in the specified direction (so that a negative value of f(e) means a flow of|f(e)|in the direction opposite the direction of e), then (11.1) simply says that the amount flowing into each vertex equals the amount flowing out. In other words, the flow is conservative. The figure below illustrates a circulation in a digraph D.


Digraph Bond Space Cycle Space Planar Embedding Maximum Forest 
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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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