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<p>In this paper, we investigate the degree of <i>h</i>-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the <i>h</i>-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this study represents the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo–Mumford regularity and the degree of the <i>h</i>-polynomial of an edge ideal achieve its maximum value, equal to the number of vertices in the graph.</p>

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Degree of h-polynomials of edge ideals

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年7月8日

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  • Jennifer Biermann,
  • Selvi Kara,
  • Augustine O’Keefe,
  • Joseph Skelton,
  • Gabriel Sosa Castillo

摘要

In this paper, we investigate the degree of h-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the h-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. To the best of our knowledge, this study represents the first investigation into the combinatorial interpretation of this algebraic invariant. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo–Mumford regularity and the degree of the h-polynomial of an edge ideal achieve its maximum value, equal to the number of vertices in the graph.