Homological invariants of Edge Ideals associated to powers of cycles
Abstract
Let Gn,m=Cn[m]̅, where Cn[m] denotes the closed mth power of the n-cycle. We study the graded Betti numbers and homological invariants of the edge ring of Gn,m in the range n≥ 3m+1. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for n≥4m+1, the graph Gn,m has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex Δ(Gn,m). We introduce a family 𝒮V(k,m) of vertex subsets characterized by their successive gaps around the cycle and show that, for W∈𝒮V(k,m), the induced subcomplex Δ[W] has the homotopy type of 𝕊1, whereas for W∉𝒮V(k,m) all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of 𝒮V(k,m), we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of Gn,m. Finally, we compute the f- and h-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case m=2 recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.
Related
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BibTeX
@misc{rather2026homological,
title = {{Homological invariants of Edge Ideals associated to powers of cycles}},
author = {Shahnawaz Ahmad Rather and S. Pirzada and M. Aijaz},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17651},
eprint = {2609.17651},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17651},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Homological invariants of Edge Ideals associated to powers of cycles
AU - Shahnawaz Ahmad Rather
AU - S. Pirzada
AU - M. Aijaz
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17651
UR - https://arxiv.org/abs/2609.17651
AB - Let G_n,m=C_n^[m]̅, where C_n^[m] denotes the closed mth power of the n-cycle. We study the graded Betti numbers and homological invariants of the edge ring of G_n,m in the range n≥ 3m+1. These graphs form a natural family for the study of edge rings whose regularity can be compared explicitly with the induced matching number. In particular, for n≥4m+1, the graph G_n,m has induced matching number one, whereas its edge ring has regularity two. Our approach is based on a characterization of the homology of the induced subcomplexes of the independence complex Δ(G_n,m). We introduce a family 𝒮_V(k,m) of vertex subsets characterized by their successive gaps around the cycle and show that, for W∈𝒮_V(k,m), the induced subcomplex Δ[W] has the homotopy type of 𝕊^1, whereas for W∉𝒮_V(k,m) all its positive-dimensional reduced homology groups vanish. Combining this characterization with Hochster's formula and an explicit enumeration of 𝒮_V(k,m), we obtain a closed formula for the graded Betti numbers in the second strand. We further determine the extremal Betti number, regularity, and projective dimension of the edge ring of G_n,m. Finally, we compute the f- and h-vectors of the independence complex and use the Hilbert series to determine the graded Betti numbers in the linear strand. The case m=2 recovers the corresponding results for complements of squares of cycles obtained in~\cite{RatherSquare}.
ER -