Arithmetical Structures on Paths With a Doubled Edge

TL;DR

This paper investigates arithmetical structures on various graph families, focusing on counting the number of such structures based on divisibility properties of vertex labelings.

Contribution

It provides new counts and insights into arithmetical structures on path graphs with doubled edges, expanding understanding of their combinatorial properties.

Findings

Counts of arithmetical structures on specific graph families

Characterization of divisibility conditions for vertex labelings

Extension of known results to new graph configurations

Abstract

An arithmetical structure on a graph is given by a labeling of the vertices which satisfies certain divisibility properties. In this note, we look at several families of graphs and attempt to give counts on the number of arithmetical structures for graphs in these families.