15
$\begingroup$

Is there a good survey on applications of Kirchhoff's circuit laws to graph theory or/and discrete geometry?

Examples:

$\endgroup$
  • 2
    $\begingroup$ I'll leave this as a comment since it's not exactly what was asked: as an outsider, I found Spectral Graph Theory by Fan R. K. Chung (AMS, 1994) very informative. $\endgroup$ – gspr May 8 '17 at 13:58
6
$\begingroup$

Possibly of help:

$\endgroup$
3
$\begingroup$

The canonical reference on all thinks Kirckhoffian is

Doyle, Peter G.; Snell, J.Laurie, Random walks and electric networks, The Carus Mathematical Monographs, 22. Washington, D. C.: The Mathematical Association of America. Distr. by John Wiley \& Sons, New York etc. XIII, 159 p. \sterling 22.00 (1984). ZBL0583.60065.

$\endgroup$
3
$\begingroup$

I really liked the discussion of electrical circuits in the recent book "Probability on Trees and Networks" by Lyons and Peres. Chapters 2, 4 and 9 seem the most relevant to what you want.

$\endgroup$
  • 3
    $\begingroup$ Adding to this: Electrical network theory is extremely important in the theory of random walks on graphs and uniform spanning trees/forests, as discussed in Lyons & Peres. Some particular things worth mentioning related to USTs: 1. Kirchoff's effective resistance formula: This expresses the probability that the UST contains a given edge in terms of the effective resistance between the endpoints. 2. Transfer Current Theorem: Allows one to compute the probability that the UST contains any given set of edges in terms of electrical quantities, due to Burton and Pemantle. $\endgroup$ – tmh May 8 '17 at 7:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.