首页|Standard monomials of 1-skeleton ideals of graphs and generalized signless Laplacians
Standard monomials of 1-skeleton ideals of graphs and generalized signless Laplacians
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NSTL
Elsevier
For a graph G on the vertex set {0, 1, ..., n}, the G -parking function ideal M-G is a monomial ideal in the polynomial ring R = K[x(1), ..., x(n)] such that the vector space dimension of R/M-G is given by the determinant of its reduced Laplacian. For any integer k, the k -skeleton ideal M-G((k)) is the subideal of M-G, where the monomial generators correspond to nonempty subsets of [n] of size at most k + 1. For a simple graph G, Dochtermann conjectured that the vector space dimension of R/M-G((1)) is bounded below by the determinant of the reduced signless Laplacian. We show that the Dochtermann conjecture holds for any (multi) graph G. More generally, we prove that this bound holds for ideals ,J(H) defined by a larger class of symmetric positive semidefinite n x n matrices H. (c) 2021 Elsevier Inc. All rights reserved.
Standard monomialsSignless LaplacianParking functionsG-PARKING FUNCTIONSTREES
NSTL
Elsevier
