In the origins of complexity theory Booth and Lueker showed that the question of whether two graphs are isomorphic or not can be reduced to the special case of chordal graphs. To prove that, they defined a transformation from graphs G to chordal graphs BL(G). The projective resolutions of the associated edge ideals I BL(G) is manageable and we investigate to what extent their Betti tables also tell non-isomorphic graphs apart. It turns out that the coefficients describing the decompositions of Betti tables into pure diagrams in Boij-Söderberg theory are much more explicit than the Betti tables themselves, and they are expressed in terms of classical statistics of the graph

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A triangulation of a polygon has an associated Stanley-Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and describe their separated models.More generally we do this for stacked simplicial complexes, in particular for stacked polytopes.

A triangulation of a polygon has an associated Stanley–Reisner ideal. We obtain a full algebraic and combinatorial understanding of these ideals and describe their separated models. More generally, we do this for stacked simplicial complexes, in particular for stacked polytopes.

A fruitful contemporary paradigm in graph theory is that almost all graphs that do not contain a certain subgraph have common structural characteristics. The "almost" is crucial, without it there is no structure. In this paper we transfer this paradigm to commutative algebra and make use of deep graph theoretic results. A key tool are the critical graphs introduced by Balogh and Butterfield.We consider edge ideals I G of graphs and their Betti numbers. The numbers of the form β i,2i+2 constitute the "main diagonal" of the Betti table. It is well known that any Betti number β i,j (I G ) below (or equivalently, to the left of) this diagonal is always zero. We identify a certain "parabola" inside the Betti table and call parabolic Betti numbers the entries of the Betti table bounded on the left by the main diagonal and on the right by this parabola. Let β i,j be a parabolic Betti number on the r-th row of the Betti table, for r ≥ 3. Our main results state that almost all graphs G with β i,j (I G ) = 0 can be partitioned into r − 2 cliques and one independent set, and in particular for almost all graphs G with β i,j (I G ) = 0 the regularity of I G is r − 1.