Arithmetical structures (Luis Garcia Puente, Colorado College)

An arithmetical structure on a finite, connected graph G without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Alternatively,  an arithmetical structure on G is a pair  of positive integer vectors (d,r) such that  Mr = 0, where M = diag(d) – A  is a square matrix whose diagonal entries are given by the vector d, and whose off-diagonal elements are given by the negative adjacency matrix of G. Arithmetical structures were first introduced by Lorenzini in 1989; matrices of the form (diag(d) – A) arise in algebraic geometry as intersection matrices of degenerating curves.  However, they also naturally appear in the context of algebraic graph theory as matrices of the form  (diag(d) – A)  generalize the Laplacian matrix of a graph.

In this talk, I will give an introduction to the topic. We will discuss some combinatorial, structural and computational aspects of arithmetical structures. In particular, we will count the number of distinct arithmetical structures on certain graph families such as path, cycle, complete and bident graphs. For paths, we will show that arithmetical structures are enumerated by the Catalan numbers. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1,n-1).  We will also discuss results about the associated critical group of an arithmetical structure, i.e.,  the cokernel of the matrix M.   This talk will be accessible to undergraduate students with some knowledge of linear algebra and discrete mathematics.