Path ideals of weighted graphs

We introduce and study the weighted r  -path ideal of a weighted graph GωGω, which is a common generalization of Conca and De Negri's r-path ideal for unweighted graphs and Paulsen and Sather-Wagstaff's edge ideal of the weighted graph. Over a field, we explicitly describe primary decompositions of these ideals, and we characterize Cohen–Macaulayness of these ideals for trees (with arbitrary r  ) and complete graphs (for r=2r=2).