On Graded Betti Numbers of Edge Rings of Certain Families of Graphs

Graphs whose edge rings have regularity two are not completely characterized. We, therefore, study the minimal graded free resolution of edge rings of certain families of graphs, such as the general barbell graph [Formula: see text] (or [Formula: see text]-barbell graph) and the complete [Formula: see text]-sunlet graph [Formula: see text], and prove that the edge ring of a barbell graph is of regularity two. We give combinatorial formulas for the graded Betti numbers in the linear strand of Stanley-Reisner rings of [Formula: see text] and [Formula: see text]. We also determine the Hilbert series of the edge ring of [Formula: see text]-barbell graph in terms of [Formula: see text]-faces of its independence complex and in terms of graded Betti numbers of the Stanley-Reisner ring of [Formula: see text]. Then we compute other graded Betti numbers of the Stanley-Reisner ring of [Formula: see text]-complexes.