Generation and properties of nut graphs

TL;DR

This paper introduces a new algorithm for generating nut graphs, characterizes their properties, and provides comprehensive data on nut graphs up to certain sizes, with implications for chemistry and electronic structure modeling.

Contribution

The paper presents a novel algorithm for exhaustive generation of nut graphs and provides the first complete enumeration of chemical nut graphs up to 22 vertices.

Findings

All chemical nut graphs have a ratio r ≥ 2.

Existence of chemical nut graphs with r = 2 for all n ≥ 9 except n=10.

Complete enumeration of nut graphs up to 13 vertices and chemical nut graphs up to 22 vertices.

Abstract

A nut graph is a graph on at least 2 vertices whose adjacency matrix has nullity 1 and for which non-trivial kernel vectors do not contain a zero. Chemical graphs are connected, with maximum degree at most three. We present a new algorithm for the exhaustive generation of non-isomorphic nut graphs. Using this algorithm, we determined all nut graphs up to 13 vertices and all chemical nut graphs up to 22 vertices. Furthermore, we determined all nut graphs among the cubic polyhedra up to 34 vertices and all nut fullerenes up to 250 vertices. Nut graphs are of interest in chemistry of conjugated systems, in models of electronic structure, radical reactivity and molecular conduction. The relevant mathematical properties of chemical nut graphs are the position of the zero eigenvalue in the graph spectrum, and the dispersion in magnitudes of kernel eigenvector entries ($r$: the ratio of maximum to minimum magnitude of entries). Statistics are gathered on these properties for all the nut graphs generated here. We also show that all chemical nut graphs have $r \ge 2$ and that there is at least one chemical nut graph with $r = 2$ for every order $n \ge 9$ (with the exception of $n = 10$).