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Cover Product and Betti Polynomial of Graphs

  Published:2015-03-25
 Printed: Jun 2015
  • Aurora Llamas,
    Departamento de Matemáticas, Cinvestav-IPN, A.P. 14--740, 07000 México D.F.
  • José Martínez-Bernal,
    Departamento de Matemáticas, Cinvestav-IPN, A.P. 14--740, 07000 México D.F.
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Abstract

For disjoint graphs $G$ and $H$, with fixed vertex covers $C(G)$ and $C(H)$, their cover product is the graph $G \circledast H$ with vertex set $V(G)\cup V(H)$ and edge set $E(G)\cup E(H)\cup\{\{i,j\}:i\in C(G), j\in C(H)\}$. We describe the graded Betti numbers of $G\circledast H$ in terms of those of $G$ and $H$. As applications we obtain: (i) For any positive integer $k$ there exists 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--Mumford regularity of $R/I(G)$ and $\mu_S(G)$ is the induced or strong matching number of $G$; (ii) The graded Betti numbers of the complement of a tree only depends upon its number of vertices; (iii) The $h$-vector of $R/I(G\circledast H)$ is described in terms of the $h$-vectors of $R/I(G)$ and $R/I(H)$. Furthermore, in a different direction, we give a recursive formula for the graded Betti numbers of chordal bipartite graphs.
Keywords: Castelnuovo--Mumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal Castelnuovo--Mumford regularity, chordal bipartite graph, edge ideal, graded Betti number, induced matching number, monomial ideal
MSC Classifications: 13D02, 05E45 show english descriptions Syzygies, resolutions, complexes
Combinatorial aspects of simplicial complexes
13D02 - Syzygies, resolutions, complexes
05E45 - Combinatorial aspects of simplicial complexes
 

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