The tau constant and the edge connectivity of a metrized graph

Details The tau constant and the edge connectivity of a metrized graph The tau constant and the edge connectivity of a metrized graph

2009-01-12

arxiv.org/abs/0901.1481v2

Mathematics

Combinatorics

Various typos are corrected. Some minor changes made to the content. However, all the results remain the same. 27 pages, 6 fig

Scientific paper

The tau constant is an important invariant of a metrized graph, and it has applications in arithmetic properties of curves. We show how the tau constant of a metrized graph changes under successive edge contractions and deletions. We discover identities which we call "contraction", "deletion", and "contraction-deletion" identities on a metrized graph. By establishing a lower bound for the tau constant in terms of the edge connectivity, we prove that Baker and Rumely's lower bound conjecture on the tau constant holds for metrized graphs with edge connectivity 5 or more. We show that proving this conjecture for 3-regular graphs is enough to prove it for all graphs.

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