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. 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.

Let G be a finite, connected graph. An arithmetical structure on G is a pair of positive integer vectors d, r such that (diag(d) − A)r = 0, where A is the adjacency matrix of G. We investigate the combinatorics of arithmetical structures on path and cycle graphs, as well as the associated critical groups (the torsion part of the cokernels of the matrices (diag(d)−A)). For paths, we prove that arithmetical structures are enumerated by the Catalan numbers, and we obtain refined enumeration results related to ballot sequences. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients 2n−1 n−1 , and we obtain refined enumeration results related to multisets. In addition, we determine the critical groups for all arithmetical structures on paths and cycles.

Abstract. This paper studies critical ideals of graphs with twin vertices, which are vertices with the same neighbors. A pair of such vertices are called replicated if they are adjacent, and duplicated, otherwise. Critical ideals of graphs having twin vertices have good properties and show regular patterns. Given a graph G = (V, E) and d ∈ Z |V | , let G d be the graph obtained from G by duplicating dv times or replicating −dv times the vertex v when dv > 0 or dv < 0, respectively. Moreover, given δ ∈ {0, 1, −1} |V | , let|V | such that dv = 0 if and only if δv = 0 and dvδv > 0 otherwise} be the set of graphs sharing the same pattern of duplication or replication of vertices. More than one half of the critical ideals of a graph in T δ (G) can be determined by the critical ideals of G. The algebraic co-rank of a graph G is the maximum integer i such that the i-th critical ideal of G is trivial. We show that the algebraic co-rank of any graph in T δ (G) is equal to the algebraic co-rank of G δ . Moreover, the algebraic co-rank can be determined by a simple evaluation of the critical ideals of G. For a large enough d ∈ Z V (G) , we show that the critical ideals of G d have similar behavior to the critical ideals of the disjoint union of G and some set {Kn v } {v∈V (G) | dv <0} of complete graphs and some set {Tn v } {v∈V (G) | dv >0} of trivial graphs. Additionally, we pose important conjectures on the distribution of the algebraic co-rank of the graphs with twins vertices. These conjectures imply that twin-free graphs have a large algebraic co-rank, meanwhile a graph having small algebraic co-rank has at least one pair of twin vertices.

Abstract. Given a graph G, an arithmetical structure on G is a pair of positive integer vectors (d, r) such that gcd(rv | v ∈ V (G)) = 1 and (diag(d) − A)r = 0, where A is the adjacency matrix of G. We describe the arithmetical structures on graph G with a cut vertex v in terms of the arithmetical structures on their blocks. More precisely, if G1, . . . , Gs are the induced subgraphs of G obtained from each of the connected components of G − v by adding the vertex v and their incident edges, then the arithmetical structures on G are in one to one correspondence with the v-rational arithmetical structures on the Gi's. We introduce the concept of rational arithmetical structure, which corresponds to an arithmetical structure where some of the integrality conditions are relaxed.