Mathematics – Combinatorics
Scientific paper
2004-07-25
Mathematics
Combinatorics
23 pages, 5 figures; corrected minor typos and altered tone slightly, updated exercises 3 and 13, updated references
Scientific paper
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 physical intuition for the concepts being dealt with. We also discuss the relation between the Laplacian on a metrized graph and the combinatorial Laplacian matrix. We introduce the``canonical measure'' on a metrized graph, which arises naturally when considering the Laplacian of the effective resistance function. Finally, we discuss a generalization of classical Fourier analysis which utilizes eigenfunctions of the Laplacian on a metrized graph. During the course of the paper, we obtain a proof of Foster's network theorem and of an intriguing series identity.
Baker Matthew
Faber Xander
No associations
LandOfFree
Metrized graphs, electrical networks, and Fourier analysis does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Metrized graphs, electrical networks, and Fourier analysis, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Metrized graphs, electrical networks, and Fourier analysis will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-629774