多重扩展完全分裂类图的边理想的同调不变量
Homological invariants of edge ideals of the multiple extended complete split-like graphs
浏览论文内容
中文总结 AI 辅助
研究图\(MECS_{b,n}^a\)的同调不变量，利用霍赫斯特公式等推导出其多种不变量的明确描述，还进行了分类、计算等工作，给出评估贝蒂数据的算法程序。
中文摘要 AI 辅助
我们研究了图\(MECS_{b,n}^a \cong \overline{K}_a \join \big(n(K_b+K_2)\big)\)，它是通过将大小为\(a\)的独立集附加到\(n\)个不相交的\(K_b+K_2\)块副本上得到的。对于\(n = 1\)，得到\(MECS_{b,1}^a\)，并恢复了[Anand, Gupta, Rather和Singh, 一些完全分裂类图的同调不变量, Beitr. Algebra Geom. (2025)]中研究的单块情况的结果。利用霍赫斯特公式、不相交变量集上的极小自由分解的张量积以及图联的贝蒂数公式，我们推导出了\(MECS_{b,n}^a\)的独立复形、独立多项式及其解析性质、希尔伯特级数、线性和二次贝蒂链、正则性、投射维数以及几个结构不变量的明确描述。我们进一步对覆盖良好和无混合的成员进行分类，计算诱导匹配数，表明该族永远不是科恩 - 麦考利的，并记录了评估贝蒂数据的算法程序。
英文摘要
We study the graphs $MECS_{b,n}^a \cong \overline{K}_a \join \big(n(K_b+K_2)\big)$, obtained by attaching an independent set of size $a$ to $n$ disjoint copies of the block $K_b+K_2$. For $n=1$, we get $MECS_{b,1}^a$, and recover the results of one-block case studied in [Anand, Gupta, Rather and Singh, Homological invariants of some complete split-like graphs, Beitr. Algebra Geom. (2025)]. Using Hochster's formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins, we derive explicit descriptions of the independence complex, independence polynomial and its analytic properties, Hilbert series, linear and quadratic Betti strands, regularity, projective dimension, and several structural invariants of $MECS_{b,n}^a$. We further classify the well-covered and unmixed members, compute induced matching numbers, show that the family is never Cohen--Macaulay, and record algorithmic procedures for evaluating the Betti data.
