Nut graphs with a given automorphism group

TL;DR

This paper demonstrates that every finite group can be realized as the automorphism group of infinitely many nut graphs, including within specific regular graph classes, expanding understanding of graph symmetry and structure.

Contribution

It proves that all finite groups can be represented as automorphism groups of nut graphs, including regular graphs with degrees 8, 12, 16, 20, and 24, with explicit constructions.

Findings

Every finite group can be realized as an automorphism group of nut graphs.

Existence of nut graphs within regular graphs of specified degrees.

Construction methods for nut graphs with prescribed automorphism groups.

Abstract

A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group can be represented as the group of automorphisms of infinitely many nut graphs. It is further shown that such nut graphs exist even within the class of regular graphs; the cases where the degree is 8, 12, 16, 20 or 24 are realised explicitly.