Standard monomials of 1-skeleton ideals of graphs and generalized signless Laplacians

Abstract

For a graph G on the vertex set {0,1,…,n}, the G-parking function ideal MG is a monomial ideal in the polynomial ring R=K[x1,…,xn] such that the vector space dimension of R/MG is given by the determinant of its reduced Laplacian. For any integer k, the k-skeleton ideal MG(k) is the subideal of MG, 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/MG(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 JH defined by a larger class of symmetric positive semidefinite n×n matrices H.