2020/06/03 by Chanchal Kumar, Kumar, Chanchal, Lather, Gargi +2
Mathematics · #05E40 #15B36 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra and Its Applications #FOS: Mathematics #Graph theory and applications
paper · pdf · doi:10.48550/arxiv.2006.02347
openalex publication_date 2020/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a (multi) graph on the vertex set V=\0,1,… ,n\ with root 0. The G-parking function ideal MG is a monomial ideal in the polynomial ring R=\mathbbK[x1,… ,xn] over a field \mathbbK such that dim\mathbb K((R)/(MG))=det(\widetildeLG), where \widetildeLG is the truncated Laplace matrix of G and det(\widetilde LG) is the determinant of \widetilde LG. In other words, standard monomials of the Artinian quotient (R)/(MG) correspond bijectively with the spanning trees of G. For 0≤ k≤ n-1, the k-skeleton ideal MG(k) of G is the monomial subideal MG(k)=⟨ mA:∅≠ A⊆[n] and |A|≤ k+1⟩ of the G-parking function ideal MG=⟨ mA : ∅ ≠ A⊆[n]⟩⊆ R. For a simple graph G, Dochtermann conjectured that dim\mathbb K(\fracRMG(1))≥det(\widetildeQG), where \widetilde QG is the truncated signless Laplace matrix of G. We show that Dochtermann conjecture holds for any (simple or multi) graph G on V.