Alexander duals of clique complexes for cyclic and chordal graphs

Determine explicit, concrete descriptions of the Alexander duals of the clique complexes associated to cyclic graphs and to chordal graphs. Provide a characterization of these Alexander dual complexes, which would enable reframing classifications of pure resolutions of edge ideals in terms of PR complexes.

Background

Section 7.2 discusses translating classifications of edge ideals with pure resolutions into the PR-complex framework. This requires describing the Alexander duals of the relevant clique complexes. The thesis supplies the Alexander dual for cross polytopes but points out that the duals for cyclic and chordal cases are not yet known.

A concrete description of these duals would link well-known graph classes (cyclic, chordal) to specific PR complexes, enabling combinatorial constructions of pure resolutions generated in fixed degrees.