We consider Stanley-Reisner rings where is the edge ideal associated to some particular classes of hypergraphs. For instance, we consider hypergraphs that are natural generalizations of graphs that are lines and cycles, and for these we compute the Betti numbers. We also generalize some known results about chordal graphs and study a weak form of shellability.
We describe some new applications of nonabelian pq-groups to construction problems in Graph Theory. The constructions include the smallest known trivalent graph of girth 17, the smallest known regular graphs of girth five for several degrees, along with four edge colorings of complete graphs that improve lower bounds on classical Ramsey numbers.
In this paper we consider some special classes of Diophantine equations connected with McFarland's and Ma's conjectures about difference sets in abelian groups and we obtain an extension of known results.
Let be a commutative ring with identity and be the set of ideals with nonzero annihilator. The strongly annihilating-ideal graph of is defined as the graph with the vertex set and two distinct vertices and are adjacent if and only if and . In this paper, the perfectness of for some classes of rings is investigated.