  This paper concerns the tau constant, which is an important invariant of a
metrized graph, and which has applications to arithmetic properties of curves.
We give several formulas for the tau constant, and show how it changes under
graph operations including deletion of an edge, contraction of an edge, and
union of graphs along one or two points. We show how the tau constant changes
when edges of a graph are replaced by arbitrary graphs. We prove Baker and
Rumely's lower bound conjecture on the tau constant for several classes of
metrized graphs.
