Regularity of symbolic and ordinary powers of weighted oriented graphs and their upper bounds

2023/08/09 by Manohar Kumar, Kumar, Manohar, Ramakrishna Nanduri +1 · 1 citation
Mathematics · Computer Science · #Graph theory and applications #Commutative Algebra and Its Applications #Graph Labeling and Dimension Problems

paper · pdf · doi:10.48550/arxiv.2308.04705

Abstract

In this paper, we compare the regularities of symbolic and ordinary powers of edge ideals of weighted oriented graphs. For any weighted oriented complete graph K_n, we show that \reg(I(K_n)^(k))≤ \reg(I(K_n)^k) for all k≥ 1. Also, we give explicit formulas for \reg(I(K_n)^(k)) and \reg(I(K_n)^k), for any k≥ 1. As a consequence, we show that \reg(I(K_n)^(k)) is eventually a linear function of k. For any weighted oriented graph D, if V⁺ are sink vertices, then we show that \reg(I(D)^(k)) ≤ \reg(I(D)^k) with k=2,3 and equality cases studied. Furthermore, we give formula for \reg(I(D)²) in terms of \reg(I(D)⁽²⁾) and regularity of certain induced subgraphs of D. Finally, we compare the regularity of symbolic powers of weighted oriented graphs D and D', where D' is obtained from D by adding a pendant.

Cited by

• Componentwise linearity of powers of edge ideals of weighted oriented graphs

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