Let $\mathcal{D}$ be a weighted oriented graph and $I(\mathcal{D})$ be its edge ideal. In this paper, we investigate the Betti numbers of $I(\mathcal{D})$ via upper-Koszul simplicial complexes, Betti splittings and the mapping cone construction. In particular, we provide recursive formulas for the Betti numbers of edge ideals of several classes of weighted oriented graphs. We also identify classes of weighted oriented graphs whose edge ideals have a unique extremal Betti number which allows us to compute the regularity and projective dimension for the identified classes. Furthermore, we characterize the structure of a weighted oriented graph $\mathcal{D}$ on $n$ vertices such that $\textrm{pdim } (R/I(\mathcal{D}))=n$ where $R=k[x_1,\ldots, x_n]$.
@article{10_37236_9887,
author = {Beata Casiday and Selvi Kara},
title = {Betti numbers of weighted oriented graphs},
journal = {The electronic journal of combinatorics},
year = {2021},
volume = {28},
number = {2},
doi = {10.37236/9887},
zbl = {1465.05071},
url = {http://geodesic.mathdoc.fr/articles/10.37236/9887/}
}
TY - JOUR
AU - Beata Casiday
AU - Selvi Kara
TI - Betti numbers of weighted oriented graphs
JO - The electronic journal of combinatorics
PY - 2021
VL - 28
IS - 2
UR - http://geodesic.mathdoc.fr/articles/10.37236/9887/
DO - 10.37236/9887
ID - 10_37236_9887
ER -
%0 Journal Article
%A Beata Casiday
%A Selvi Kara
%T Betti numbers of weighted oriented graphs
%J The electronic journal of combinatorics
%D 2021
%V 28
%N 2
%U http://geodesic.mathdoc.fr/articles/10.37236/9887/
%R 10.37236/9887
%F 10_37236_9887
Beata Casiday; Selvi Kara. Betti numbers of weighted oriented graphs. The electronic journal of combinatorics, Tome 28 (2021) no. 2. doi: 10.37236/9887
