Arithmetical Structures on Coconut Trees
Alexander Diaz-Lopez, Brian Ha, Pamela E. Harris, Jonathan Rogers,, Theo Koss, Dorian Smith

TL;DR
This paper extends the enumeration of arithmetical structures from specific coconut tree graphs to a broader class, providing a comprehensive characterization of these structures on all coconut trees.
Contribution
It generalizes previous enumeration results to all coconut tree graphs and characterizes smooth arithmetical structures with leaf node assignments.
Findings
Enumeration of arithmetical structures on all coconut trees
Characterization of smooth arithmetical structures
Extension of previous results on bidents
Abstract
If G is a finite connected graph, then an arithmetical structure on is a pair of vectors with positive integer entries such that , where is the adjacency matrix of and the entries of have no common factor other than . In this paper, we generalize the result of Archer, Bishop, Diaz-Lopez, Garc\'ia Puente, Glass, and Louwsma on enumerating arithmetical structures on bidents (also called coconut tree graphs ) to all coconut tree graphs which consists of a path on vertices to which we append 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.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Graph theory and applications · Advanced Graph Theory Research
