Algebraic study on Cameron--Walker graphs
Algebraic study on Cameron--Walker graphs
复制标题
卡梅伦--沃克图的代数研究
DOI：
10.1016/j.jalgebra.2014.07.037
复制
发表时间：
2015
影响因子：
0.9
通讯作者：
Augustine B. O'Keefe
中科院分区：
文献类型：
--
作者：
Takayuki Hibi;Akihiro Higashitani;Kyouko Kimura;Augustine B. O'Keefe
设G是[n]上的有限简单图，I（G）<$S是G的边理想，其中S= K [x1，...，xn]是域K上的多项式环.设m（G）表示G的匹配的最大尺寸，im（G）表示G的诱导匹配的最大尺寸.已知im（G）≤ reg（S/I（G））≤ m（G），其中reg（S/I（G））是S/I（G）的Castelnuovo-Mumford正则性.卡梅隆和步行者成功地对满足im（G）= m（G）的有限连通简单图G进行了分类.我们称有限连通简单图G是Cameron-Walker图，如果i m（G）= m（G）且G既不是星星也不是星星三角形.本文从交换代数的角度研究Cameron-Walker图。首先，我们证明了Cameron-Walker图G是非混合图当且仅当G是Cohen-Macaulay图，并对所有的Cohen-Macaulay Cameron-Walker图进行了分类。其次，证明了不存在Gorenstein Cameron-Walker图。最后证明了每个Cameron-Walker图都是序列Cohen-Macaulay图。
Let G be a finite simple graph on [n] and I (G)⊂ S the edge ideal of G, where S= K [x 1,…, x n] is the polynomial ring over a field K. Let m (G) denote the maximum size of matchings of G and i m (G) that of induced matchings of G. It is known that i m (G)≤ reg (S/I (G))≤ m (G), where reg (S/I (G)) is the Castelnuovo–Mumford regularity of S/I (G). Cameron and Walker succeeded in classifying the finite connected simple graphs G with i m (G)= m (G). We say that a finite connected simple graph G is a Cameron–Walker graph if i m (G)= m (G) and if G is neither a star nor a star triangle. In the present paper, we study Cameron–Walker graphs from a viewpoint of commutative algebra. First, we prove that a Cameron–Walker graph G is unmixed if and only if G is Cohen–Macaulay and classify all Cohen–Macaulay Cameron–Walker graphs. Second, we prove that there is no Gorenstein Cameron–Walker graph. Finally, we prove that every Cameron–Walker graph is sequentially Cohen–Macaulay.