Classification of quartic bicirculant nut graphs

TL;DR

This paper classifies quartic bicirculant nut graphs by analyzing their structural properties across four classes of such graphs, advancing understanding of their spectral and automorphic characteristics.

Contribution

It provides a complete classification of quartic bicirculant nut graphs, a previously unexplored intersection of graph spectral theory and automorphism properties.

Findings

Identified all quartic bicirculant nut graphs within four classes

Characterized spectral properties of these graphs

Established criteria for nut graph classification

Abstract

A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four classes of connected quartic bicirculant graphs. We classify the quartic bicirculant graphs that are nut graphs by investigating properties of each of these four classes.