On the jacobian module

associated to a graph

Author:
Aron Simis

Journal:
Proc. Amer. Math. Soc. **126** (1998), 989-997

MSC (1991):
Primary 13H10; Secondary 13D40, 13D45, 13H15

DOI:
https://doi.org/10.1090/S0002-9939-98-04180-X

MathSciNet review:
1425139

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the *jacobian module* of a set of squarefree monomials of degree corresponding to the edges of a connected bipartite graph . We show that for such a graph the number of its primitive cycles (i.e., cycles whose chords are not edges of ) is the second Betti number in a minimal resolution of the corresponding jacobian module. A byproduct is a graph theoretic criterion for the subalgebra to be a complete intersection.

**1.**F. Harary,*Graph Theory*, Addison-Wesley Publishing Co., Reading, Mass., 1969. MR**41:1566****2.**L. R. Doering and T. Gunston, Algebras arising from bipartite planar graphs, Comm. Algebra**24**(1996), 3589-3598. CMP**96:17****3.**A. Simis, W. V. Vasconcelos and R. Villarreal, On the ideal theory of graphs, J. Algebra**167(2)**(1994), 389-416. MR**95e:13002****4.**R. Villarreal, Cohen-Macaulay graphs, Manuscripta Math.**66**(1990), 277-293. MR**91b:13031****5.**R. Villarreal, Rees algebras of edge-ideals, Comm. Algebra**23**(1995), 3513-3524. MR**96e:13005**

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Additional Information

**Aron Simis**

Email:
aron@ufba.br

DOI:
https://doi.org/10.1090/S0002-9939-98-04180-X

Received by editor(s):
June 1, 1996

Received by editor(s) in revised form:
September 27, 1996

Additional Notes:
The author was partially supported by CNPq, Brazil.

Communicated by:
Wolmer V. Vasconcelos

Article copyright:
© Copyright 1998
American Mathematical Society