Theodore Koss, “Counting Arithmetical Structures on Coconut Trees”
Mentor: Pamela Harris, Mathematical Sciences
Poster #111

If G is a finite and connected graph, then an arithmetical structure on G is a pair of vectors (d, r) with positive integer entries such that (diag(d) – A)r = 0, where A is the adjacency matrix of G and the entries of r have no common factor. An equivalent definition is a labelling of the vertices of a graph such that the value at a vertex divides the sum of its neighbors’ values, and all the values have no common factor. In this paper, we generalize the result of Archer, Bishop, Diaz-Lopez, Garcia Puente, Glass and Louwsma on enumerating arithmetical structures on bidents (also called coconut tree graphs CT(p, 2)) to all coconut tree graphs CT(p, s) with p and s being any positive integers. We also give a characterization of fully smooth arithmetical structures on coconut trees when given number assignments to the leaf nodes.