Progress:
All of the students wrote MAPLE code contributing to this endeavor. See in particular Phil's MAPLE package.
Progress:
Phil discovered using MAPLE that a previously conjectured lower bound for the Tau constant (1/16) is wrong; he found numerous examples of graphs with smaller values of Tau.
Crystal worked out, using Lagrange multipliers, for certain "banana graphs" that Tau is minimized when all of the edge lengths are equal.
Progress:
Xander proved (see his writeup) that the eigenvalues of the discrete Laplacian converge to the continuous ones under suitable hypotheses.
Phil, Kinsey, Kevin, and Rommel gathered evidence for, and formulated a conjecture that for the dx measure, the eigenvalues of the discrete Laplacian increase monotonically toward the eigenvalues of the continuous Laplacian (when appropriately scaled).
Kinsey and Kevin discovered examples of eigenvalues of the discrete Laplacian (with respect to the canonical measure) which don't always monotonically increase toward the continuous value. This was a bit surprising, since for the dx measure they always seem to monotonically increase.
Using least-squares fits, Kinsey found that for the dx measure, the difference between the eigenvalues in the discrete case and the continuous case seems to go to zero at a rate proportional to n-3/2, if the edges of the graph are subdivided into n equal pieces.
Progress:
Maxim and Rommel worked out (and tabulated in spreadsheets) lots of examples of continuous eigenvalues of the Laplacian for various graphs. Kinsey and Kevin worked out the discrete analogues.
Collecting all their data together, Phil, Kinsey, Kevin, and Rommel compiled biographies for a large number of example graphs, including "stars", "flowers", "bananas", the edge graphs of the five platonic solids, various bipartite graphs, and the Peterson graph. The data for each graph included its canonical measure, the value of the tau constant, and lists of the first several eigenvalues in the discrete and continuous case (including their multiplicities).
Xander worked out the details of an example of J.P. Roth which shows that "you can't hear the shape of a metrized graph".
Progress:
Xander and Phil proved that there is a discrete version of the canonical measure, and worked out many properties it has analogous to the continuous case. See the notes written by Phil. Phil used the discrete canonical measure for computational purposes in some of his programs.
Progress:
Jake worked out some nice connections between the canonical measure and spanning trees. Specifically, he obtained an alternate proof using Kirchoff's Matrix Tree Theorem of an interesting identity involving effective resistances that drops out of certain facts about the canonical measure. He also reformulated certain quantities related to Tau as counting certain kinds of spanning trees.