Distance ideals of graphs

TL;DR

This paper introduces the concept of distance ideals in graphs, generalizing the Smith normal form and spectra of the distance matrix, and classifies graphs with at most one trivial distance ideal.

Contribution

It proposes a new concept of distance ideals for graphs and provides a classification for graphs with limited trivial distance ideals.

Findings

Classification of graphs with at most one trivial distance ideal

Generalization of Smith normal form and spectra of the distance matrix

Introduction of the concept of distance ideals

Abstract

We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial distance ideal.