发表机构
Süleyman Demirel University; Indian Institute of Technology, Bhilai（苏莱曼·德米雷尔大学; 印度理工学院比莱分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究无孤立顶点图的闭邻域超图射影维数与扩展二部二重覆盖的正则性关系，证明一般上界，并在多类图上建立精确等式，方法结合覆盖、支配、匹配参数与同调工具。
AI 中文摘要
设 $G$ 是一个无孤立顶点的有限简单图。我们研究闭邻域超图 $\mathcal{N}[G]$ 的射影维数，及其与 $G$ 的扩展二部二重覆盖 $\mathfrak{B}_e(G)$ 的 Castelnuovo-Mumford 正则性之间的关系。我们为所有图建立了一般上界 $\operatorname{prod-dim} (\mathcal{N}[G]) \leq \operatorname{reg}(\mathfrak{B}_e(G))$。此外，我们证明当 $G$ 属于若干著名图类时，精确等式 $\operatorname{prod-dim} (\mathcal{N}[G]) = \operatorname{reg}(\mathfrak{B}_e(G)) =\alpha(G)$ 成立，这些图类包括 König-Egerváry 图（包含所有二部图）、cographs、co-chordal 图、弦图和可比图，其中 $\alpha(G)$ 表示独立数。我们的证明方法依赖于通过覆盖、支配和匹配参数将代数不变量与图的底层组合结构联系起来，并结合同调工具的使用。
英文摘要
Let $G$ be a finite and simple graph without isolated vertices. We investigate the projective dimension of the closed neighborhood hypergraph $\mathcal{N}[G]$ and its relationship with the Castelnuovo-Mumford regularity of the extended bipartite double cover $\mathfrak{B}_e(G)$ of $G$. We establish the general upper bound $\operatorname{prod-dim} (\mathcal{N}[G]) \leq \operatorname{reg}(\mathfrak{B}_e(G))$ for all graphs. Furthermore, we prove that the exact equalities $\operatorname{prod-dim} (\mathcal{N}[G]) = \operatorname{reg}(\mathfrak{B}_e(G)) =α(G)$ hold when $G$ belongs to several prominent graph classes, including König-Egerváry (contains all bipartite graphs), cographs, co-chordal, chordal and comparability graphs, where $α(G)$ denotes the independence number. Our method of proofs relies on connecting algebraic invariants to the underlying combinatorial structure of graphs through covering, domination and matching parameters, together with the use of homology tools.
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