Lifting Constructions of Strongly Regular Cayley Graphs

TL;DR

This paper introduces two novel methods for constructing strongly regular Cayley graphs using lifting techniques, one based on cyclotomic graphs and the other on quadratic forms, leading to new association schemes.

Contribution

It presents two new lifting constructions for strongly regular Cayley graphs, generalizing existing methods and establishing a recursive relationship between the constructions.

Findings

Constructed new strongly regular Cayley graphs using cyclotomic and quadratic form methods.

Established a recursive relationship between the two constructions.

Derived association schemes from the second construction.

Abstract

We give two "lifting" constructions of strongly regular Cayley graphs. In the first construction we "lift" a cyclotomic strongly regular graph by using a subdifference set of the Singer difference set. The second construction uses quadratic forms over finite fields and it is a common generalization of the construction of the affine polar graphs \cite{CK86} and a construction of strongly regular Cayley graphs given in \cite{FWXY}. The two constructions are related in the following way: The second construction can be viewed as a recursive construction, and the strongly regular Cayley graphs obtained from the first construction can serve as starters for the second construction. We also obtain association schemes from the second construction.