Cover Product and Betti Polynomial of Graphs

Canadian mathematical bulletin

Aurora Llamas¹  José Martínez-Bernal¹ 

[1] Departamento de Matemáticas, Cinvestav-IPN, A.P. 14--740, 07000 México D.F.

关键词: Castelnuovo--Mumford regularity;    chordal bipartite graph;    edge ideal;    graded Betti number;    induced matching number;    monomial ideal;   

DOI  :  10.4153/CMB-2015-013-3

学科分类：数学（综合）

来源: University of Toronto Press * Journals Division

【 摘 要 】

For disjoint graphs $G$ and $H$, with fixedvertex covers$C(G)$ and $C(H)$, their cover product is the graph $GcircledastH$ with vertex set$V(G)cup V(H)$ and edge set $E(G)cup E(H)cup{{i,j}:iinC(G), jinC(H)}$. We describe the graded Betti numbers of $GcircledastH$ in terms of those of$G$ and $H$. As applications we obtain: (i) For any positiveinteger $k$ thereexists a connected bipartite graph $G$ such that $operatorname{reg}R/I(G)=mu_S(G)+k$, where,$I(G)$ denotes the edge ideal of $G$, $operatorname{reg} R/I(G)$is the Castelnuovo--Mumfordregularity of $R/I(G)$ and $mu_S(G)$ is the induced or strongmatching number of$G$; (ii) The graded Betti numbers of the complement of a treeonly depends uponits number of vertices; (iii) The $h$-vector of $R/I(GcircledastH)$ is described interms of the $h$-vectors of $R/I(G)$ and $R/I(H)$. Furthermore,in a differentdirection, we give a recursive formula for the graded Betti numbersof chordalbipartite graphs.

【 授权许可】

Unknown