For a graph G and any of its splitting graphs G', the paper proves inequalities such as reg(I(G)) <= reg(I(G')) and proj dim(I(G)) <= proj dim(I(G')) for special splittings and many graph classes.
Jacques, Betti numbers of graph ideals, Thesis (2004)
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The edge ideal of a graph and its splitting graphs
For a graph G and any of its splitting graphs G', the paper proves inequalities such as reg(I(G)) <= reg(I(G')) and proj dim(I(G)) <= proj dim(I(G')) for special splittings and many graph classes.