Is there a good survey on applications of Kirchhoff's circuit laws to graph theory or/and discrete geometry?
Examples:
$\begingroup$
$\endgroup$
1
asked May 8, 2017 at 1:56
-
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. $\endgroup$– gsprCommented May 8, 2017 at 13:58
Add a comment
|
3 Answers
$\begingroup$
$\endgroup$
Possibly of help:
Pavlo Pylyavskyy, UMN Math 8680 Spring 2017, scribe notes (by students). (Scroll down to the attachments section for the scribe notes. The papers are less closely related to the electrical networks theme.)
$\begingroup$
$\endgroup$
1
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.
-
4$\begingroup$ 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. $\endgroup$– tmhCommented May 8, 2017 at 7:19
$\begingroup$
$\endgroup$
2
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.
answered May 8, 2017 at 3:02
-
1$\begingroup$ It does not seem to include things like counting trees... $\endgroup$ Commented May 8, 2017 at 3:58
-
2$\begingroup$ Doyle and Snell is available online at math.dartmouth.edu/~doyle/docs/walks/walks.pdf $\endgroup$ Commented May 8, 2017 at 14:44