Distance-regular Cayley graphs with least eigenvalue $-2$

TL;DR

This paper classifies distance-regular Cayley graphs with least eigenvalue -2 and diameter up to three, identifying specific graph families and analyzing their structural properties.

Contribution

It provides a complete classification of such graphs, including sporadic cases, lattice graphs, certain triangular graphs, and line graphs of incidence graphs of specific projective planes.

Findings

Classification of distance-regular Cayley graphs with eigenvalue -2

Identification of lattice, triangular, and line graph families

Structural insights into Cayley line graphs of generalized polygons

Abstract

We classify the distance-regular Cayley graphs with least eigenvalue $-2$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain projective planes. In addition, we classify the possible connection sets for the lattice graphs and obtain some results on the structure of distance-regular Cayley line graphs of incidence graphs of generalized polygons.