On a Class of Semigroup Graphs

Let G = Γ(S) be a semigroup graph, i.e., a zero-divisor graph of a semigroup S with zero element 0. For any adjacent vertices x, y in G, denote C(x,y) = {z∈V(G) | N (z) = {x,y}}. Assume that in G there exist two adjacent vertices x, y, a vertex s∈C(x,y) and a vertex z such that d (s,z) = 3. This paper studies algebraic properties of S with such graphs G = Γ(S), giving some sub-semigroups and ideals of S. It constructs some classes of such semigroup graphs and classifies all semigroup graphs with the property in two cases.

A Construction of Sequentially Cohen-Macaulay Graphs

For every simple graph G,a class of multiple clique cluster-whiskered graphs G^(eπm)is introduced,and it is shown that all such graphs are vertex decomposable;thus,the independence simplicial complex Ind G^(eπm)is sequentially Cohen-Macaulay.The properties of the graphs G^(eπm)and G^(π)constructed by Cook and Nagel are studied,including the enumeration of facets of the complex Ind G^(π)and the calculation of Betti numbers of the cover ideal Ic(G^(eπm).We also prove that the complex△=IndH is strongly shellable and pure for either a Boolean graph H=Bn or the full clique-whiskered graph H=G^(W)of C,which is obtained by adding a whisker to each vertex of G.This implies that both the facet ideal I(△)and the cover ideal Ic(H)have linear quotients.

Edge Ideals with Regularity No More than Three

We prove that if G is a gap-free and chair-free simple graph,then the regularity of the edge ideal of G is no more than 3.If G is a gap-free and P4-free graph,then it is a chair-free graph;furthermore,the complement of G is chordal,and thus the regularity of G is 2.

Finite Rings Whose Graphs Have Clique Number Less than Five

Let R be a commutative ring and Γ(R)be its zero-divisor graph.We completely determine the structure of all finite commutative rings whose zero-divisor graphs have clique number one,two,or three.Furthermore,if R■R1×R2×…×Rn(each Ri is local for i=1,2,3,...,n),we also give algebraic characterizations of the ring R when the clique number of Γ(R)is four.

Graph Properties and Stratified Presentations of Partially Ordered Sets

In this paper, we introduce some new definitions such as the U＊L＊ condition to describe the zero-divisor graph G = F（P） of a poser P, and give a new and quick proof to a main result in [2, 4]. By deleting a typical vertex with least degree, we provide an algorithm for finding a maximum clique of a finite graph G. We study some properties of the zero-divisor graphs of posets concerning diameters and girths. We also provide stratified presentations of posets.

Zero divisors and prime elements of bounded semirings

A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A bounded semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of zero divisors and prime elements of a bounded semiring are studied. In particular, it is proved that under some mild assumption, the set Z（A） of nonzero zero divisors of A is A ／ {0, 1}, and each prime element of A is a maximal element. For a bounded semiring A with Z（A） = A ／ {0, 1}, it is proved that A has finitely many maximal elements if ACC holds either for elements of A or for principal annihilating ideals of A. As an application of prime elements, we show that the structure of a bounded semiring A is completely determined by the structure of integral bounded semirings if either ｜Z（A）｜ = 1 or ｜Z（A）｜ -- 2 and Z（A）2 ≠ 0. Applications to the ideal structure of commutative rings are also considered. In particular, when R has a finite number of ideals, it is shown that the chain complex of the poset I（R） is pure and shellable, where I（R） consists of all ideals of R.

Artinian Local Rings Whose Annihilating-ideal Graphs Are Star Graphs

In this paper, a necessary and sufficient condition is given for a commutative Artinian local ring whose annihilating-ideal graph is a star graph. Also, a complete char- acterization is established for a finite local ring whose annihilating-ideal graph is a star graph.

On Finite Local Rings with Clique Number Four

We study the algebraic structure of rings R whose zero-divisor graph T(R)has clique number four.Furthermore,we give complete characterizations of all the finite commutative local rings with clique number 4.

On Graded Betti Numbers of a Class of Graphs

In this paper, Betti numbers are evaluated for several classes of graphs whose complements are bipartite graphs. Relations are established for the general case, and counting formulae are given in several particular cases, including the union of several mutually disjoint complete graphs.