  We study weighted graphs and their "edge ideals" which are ideals in
polynomial rings that are defined in terms of the graphs. We provide
combinatorial descriptions of m-irreducible decompositions for the edge ideal
of a weighted graph in terms of the combinatorics of "weighted vertex covers".
We use these, for instance, to say when these ideals are m-unmixed. We
explicitly describe which weighted cycles, suspensions, and trees are unmixed
and which ones are Cohen-Macaulay, and we prove that all weighted complete
graphs are Cohen-Macaulay.
