Critical groups of arithmetical structures on star graphs and complete graphs

TL;DR

This paper characterizes the critical groups associated with arithmetical structures on star and complete graphs, providing formulas and classifications for the finite abelian groups that can arise.

Contribution

It introduces methods to compute critical groups from arithmetical structures on star and complete graphs and classifies possible groups.

Findings

Explicit formulas for critical groups on star and complete graphs

Classification of finite abelian groups as critical groups

Enhanced understanding of arithmetical structures on these graphs

Abstract

An arithmetical structure on a finite, connected graph without loops is an assignment of positive integers to the vertices that satisfies certain conditions. Associated to each of these is a finite abelian group known as its critical group. We show how to determine the critical group of an arithmetical structure on a star graph or complete graph in terms of the entries of the arithmetical structure. We use this to investigate which finite abelian groups can occur as critical groups of arithmetical structures on these graphs.