# Digraphs with at most one trivial critical ideal

## TL;DR

This paper characterizes digraphs with at most one trivial critical ideal, linking properties of their critical groups, Smith groups, and characteristic polynomials, thus extending understanding of algebraic invariants of digraphs.

## Contribution

It provides a complete characterization of digraphs with at most one trivial critical ideal, connecting critical ideals with critical and Smith groups.

## Key findings

- Characterization of digraphs with at most one trivial critical ideal
- Implications for critical groups with a single invariant factor of one
- Implications for Smith groups with a single invariant factor of one

## Abstract

Critical ideals generalize the critical group, Smith group and the characteristic polynomials of the adjacency and Laplacian matrices of a graph. We give a complete characterization of the digraphs with at most one trivial critical ideal. Which implies the characterizations of the digraphs whose critical group has one invariant factor equal to one, and the digraphs whose Smith group has one invariant factor equal to one.