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.
This paper studies the coquasitriangularity of Hopf superalgebras in the framework of Hopf superquivers. We give a complete description of coquasitriangular structures that concentrate at degree 0 on Hopf superquivers. We obtain a classification of finite-dimensional coquasitriangular pointed Hopf superalgebras.
In this paper, two Frattini embedding theorems on nilpotent groups are proved. First, let [Formula: see text] be a finitely generated nilpotent group whose torsion subgroup is a finite [Formula: see text]-group, and then there exists a finitely generated nilpotent group [Formula: see text] whose torsion subgroup is also a finite [Formula: see text]-group such that [Formula: see text]. Second, let [Formula: see text] be a nilpotent Cernikov [Formula: see text]-group for a finite set [Formula: see text] of primes, and then there exists a nilpotent Cernikov [Formula: see text]-group [Formula: see text] such that [Formula: see text] and [Formula: see text].
Let [Formula: see text] be a 2-torsion-free commutative ring with unity, [Formula: see text] a locally finite pre-ordered set and [Formula: see text] the incidence algebra of [Formula: see text] over [Formula: see text]. We prove that under certain conditions every generalized Lie derivation and every [Formula: see text]-Lie derivation (with a linear map [Formula: see text]) of [Formula: see text] is the sum of a left multiplication map, a derivation and a central map which vanishes on all commutators of [Formula: see text].
Let [Formula: see text] be a finite permutation group acting on a set [Formula: see text]. An ordered sequence [Formula: see text] of elements of [Formula: see text] is an irredundant base for [Formula: see text] if the pointwise stabilizer of the sequence is trivial and no point is fixed by the stabilizer of its predecessors. We show that any interval of natural numbers can be realized as the set of cardinalities of irredundant bases for some finite primitive group.
