We introduce some determinantal ideals of the generalized Laplacian matrix
associated to a digraph G, that we call critical ideals of G. Critical ideals
generalize the critical group and the characteristic polynomials of the
adjacency and Laplacian matrices of a digraph. The main results of this article
are the determination of some minimal generator sets and the reduced Grobner
basis for the critical ideals of the complete graphs, the cycles and the paths.
Also, we establish a bound between the number of trivial critical ideals and
the stability and clique numbers of a graph.Comment: 23 pages. Some changes over the previous version. Accepted in Linear
algebra and its Application
Abstract. The critical ideals of a graph are the determinantal ideals of the generalized Laplacian matrix associated to a graph. A basic property of the critical ideals of graphs asserts that the graphs with at most k trivial critical ideals, Γ ≤k , are closed under induced subgraphs. In this article we find the set of minimal forbidden subgraphs for Γ ≤2 , and we use this forbidden subgraphs to get a classification of the graphs in Γ ≤2 . As a consequence we give a classification of the simple graphs whose critical group has two invariant factors equal to one. At the end of this article we give two infinite families of forbidden subgraphs.
The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles. A thick cycle graph is a cycle where multi-edges are permitted. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graph's edge multiplicities.
Let G be a simple graph with V G = n and no isolated vertices. Let be its stability number. We study invariants of the edge-ring of G that can be interpreted as invariants of G . If G has a cover by maximum stable sets we are able to prove the inequality ≤ n 2 . As a byproduct we prove that if G is vertex-critical, then ≤ n − A /2, where A is the intersection of all the minimum vertex covers of G . We estimate the smallest number of vertices in any maximal stable set of G to obtain a bound for the depth of the edge-ring of G .
Abstract. Arithmetical structures on a graph were introduced by Lorenzini in [9] as some intersection matrices that arise in the study of degenerating curves in algebraic geometry. In this article we study these arithmetical structures, in particular we are interested in the arithmetical structures on complete graphs, paths, and cycles. We begin by looking at the arithmetical structures on a multidigraph from the general perspective of M -matrices. As an application, we recover the result of Lorenzini about the finiteness of the number of arithmetical structures on a graph. We give a description on the arithmetical structures on the graph obtained by merging and splitting a vertex of a graph in terms of its arithmetical structures. On the other hand, we give a description of the arithmetical structures on the clique-star transform of a graph, which generalizes the subdivision of a graph. As an application of this result we obtain an explicit description of all the arithmetical structures on the paths and cycles and we show that the number of the arithmetical structures on a path is a Catalan number.
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