2024/06/17 by Alexander Diaz-Lopez, Brian Ha, Diaz-Lopez, Alexander +9
Computer Science · Mathematics · #Graph Labeling and Dimension Problems #Graph theory and applications #Advanced Graph Theory Research
paper · pdf · doi:10.48550/arxiv.2406.11183
If G is a finite 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 other than 1. In this paper, we generalize the result of Archer, Bishop, Diaz-Lopez, García Puente, Glass, and Louwsma on enumerating arithmetical structures on bidents (also called coconut tree graphs \CTp2) to all coconut tree graphs \CTps which consists of a path on p>0 vertices to which we append s>0 leaves to the right most vertex on the path. We also give a characterization of smooth arithmetical structures on coconut trees when given number assignments to the leaf nodes.