Homological properties of the edge ideals for trees with small diameter

Articles in Press

Document Type : Original paper

Authors

1 School of Mathematics and Statistics, Shandong University of Technology, Zibo 255049, China

2 School of Mechanical Engineering, Shandong University of Technology, Zibo 255049, China

3 Department of Mathematics and Applications, University ‘Federico II’, Naples I-80125, Italy

Abstract

Let $T$ be a tree of diameter at most $5$. We investigate homological invariants of its edge ideal, including the projective dimension, the Castelnuovo–Mumford regularity, and the graded Betti numbers. For trees of diameter at most $3$, all nonzero Betti numbers lie on the linear strand. For trees of diameter~$4$ and~$5$, we determine the regularity and the projective dimension explicitly. In the case of caterpillar trees of diameter~$4$, we compute all graded Betti numbers and provide an explicit formula relating them to the $f$-vector of the independence complex. Our results refine the combinatorial description of Betti numbers for forests and highlight structural features of trees with small diameter.

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References

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Communications in Combinatorics and Optimization

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Available Online from 10 May 2026

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