## Abstract

For an edge ideal I(G) of a simple graph (Formula presented.) we study the (Formula presented.) -graded Betti numbers that appear in the linear strand of the minimal free resolution of (Formula presented.) where (Formula presented.) is the comaximal graph of the integral modulo ring (Formula presented.). We show that the extremal Betti number of (Formula presented.) is (Formula presented.) where (Formula presented.) is the Euler’s totient function and thereby we obtain a large class of edge ideals with even extremal Betti numbers. We find the regularity (Castelnuovo-Mumford) and the projective dimension of these ideals. Moreover, we exhibit explicit formulae that determine all the (Formula presented.) -graded Betti numbers in the linear strand of the minimal free resolution of (Formula presented.) for certain values of (Formula presented.).

Original language | English |
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Pages (from-to) | 1486-1500 |

Number of pages | 15 |

Journal | Communications in Algebra |

Volume | 52 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2024 |

## Keywords

- Betti numbers
- Euler’s totient function
- comaximal graphs
- edge ideals
- integers modulo ring
- linear strand
- projective dimension
- regularity

## ASJC Scopus subject areas

- Algebra and Number Theory