Algebraic properties of spanning simplicial complexes (Q1683394)

A significant line of research in combinatorial commutative algebra is finding classes of simplicial complexes with desired properties, such as vertex decomposability, shellability, and Cohen-Macaulaynes. Simplicial complexes may be constructed from graphs or multigraphs, and the combinatorial and algebraic properties of the simplicial complexes may be comparied to the properties of the graph. The spanning simplicial complex is one way of making this association: given a multigraph \(G\), the facets of the spanning simplicial complex, denoted \(\Delta_s(G)\), is the complex whose facets are the edge sets of all spanning trees of \(G\) [Definition 2.1]. In this paper, the authors study algebraic and homological properties associated to the spanning simplicial complex of a multigraph and to its Stanley-Reisner ring. The authors prove that the spanning simplicial complex \(\Delta_s(G)\) of a multigraph \(G = (V(G), E(G))\) is vertex decomposable, and therefore shellable [Theorem 2.7]. In order to prove this result by induction, they first show that the link of an edge \(e\) in \(\Delta_s(G)\) is equal to the spanning simplicial complex of the contraction \(G/e\) and that the deletion of an edge \(e\) in \(\Delta_s(G)\) is equal to the spanning simplicial complex of the deletion \(G-e\) [Lemma 2.6]. Since \(\Delta_s(G)\) is a pure simplicial complex of dimension \(V(G) - 2\) [Remark 2.4] and all pure shellable complex are Cohen-Macaulay, it follows that every spanning simplicial complex of a multigraph are Cohen-Macaulay [Corollary 2.8]. In Corollary 2.9, it is shown that the Stanley-Reisner ideal of \(\Delta_s(G)\) is generated by monomials \(x_{i_1} \cdots x_{i_k}\) where \(\{ x_{i_1}, \dots, x_{i_k}\}\) is in the edge set of a cycle of \(G\); given the previous results, this provides an explicit description of the generators of a class of Cohen-Macaulay rings. The authors also use the vertex decomposability of \(\Delta_s(G)\) to demonstrate that the Stanley-Reisner ideal of the Alexander dual simplicial complex \(\Delta_s(G)^{\vee}\) has linear quotients and a linear resolution [Corollary 2.13]. The final section of the paper investigates relationships between the Stanley-Reisner ring of the spanning simplicial complex \(\Delta_s(G)\) and the multigraph \(G\) in certain cases. For a connected multigraph \(G\), it is shown that the dimension of the Stanley-Reisner ring is equal to \(|V(G)| -1\) and that the projective dimension of the Stanley Reisner ring is equal to \(|E(G) - |V(G)| +1\), which is also equal to the circuit rank of \(G\) [Theorem 3.2]. Finally, for special multigraphs, both the projective dimension and the Castelnuovo-Mumford regularity of the Stanley-Reisner ring of the spanning simplicial complex are described in terms of properties of \(G\) [Corollaries 3.3 and 3.4].