Arithmetical structures on graphs

Abstract

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.

Arithmetical structures on graphs with connectivity one

“…Lorenzini proved in [6] that if G is a simple connected graph, then there are a finite number of arithmetical structures. In [4] this result was generalized to strongly connected multidigraphs. On the other hand, given an arithmetical structure (d, r) on G, letbe its critical group, which generalizes the concept of the critical group of G introduced in [1].…”

“…In [2] there was described in detail the combinatorics of the arithmetical structures on the path and cycle graphs. Also, in [4] there was introduced and studied the arithmetical structures in the general setting of M -matrices and in particular the arithmetical structures on complete graphs, paths, and cycles were studied. Moreover, there was described a subset of the arithmetical structures on the cliquestar transformation of a graph G in dependence on the arithmetical structures on G. Apart from the arithmetical structures on the path and cycle, in general the description of the arithmetical structures on a graph is a very difficult problem.…”

Counting arithmetical structures on paths and cycles

“…In brief, the path P n and the cycle C n on n vertices satisfy | Arith(P n )| = C n−1 = 1 n 2n − 2 n − 1 , | Arith(C n )| = 2n − 1 n − 1 = (2n − 1)C n−1 (Theorems 3 and 30, respectively). These results were announced in [CV18].…”

Minimal free resolutions of lattice ideals of digraphs

“…Moreover, L is an ICB matrix if and only if G L is strongly connected (see [8], for a recent related work). Indeed, first remark that a permutation of L is just a re-enumeration of the vertex set of G and a corresponding relabelling of the directed arcs.…”