Cyclic arcs of Singer type and strongly regular Cayley graphs over finite fields

TL;DR

This paper constructs infinite families of strongly regular Cayley graphs over finite fields using cyclic arcs of Singer type, extending previous results to new parameter sets under specific algebraic conditions.

Contribution

It proves the existence of infinitely many strongly regular Cayley graphs with certain parameters for new cases where M divides h^2+h+1 and the order of 2 in (Z/MZ)^× is odd.

Findings

Existence of strongly regular Cayley graphs with negative Latin square type parameters.

Extension of known cases to infinitely many primes q under algebraic divisibility and order conditions.

New construction methods for Cayley graphs over finite fields.

Abstract

In \cite{M18}, the first author gave a construction of strongly regular Cayley graphs on the additive group of finite fields by using three-valued Gauss periods. In particular, together with the result in \cite{BLMX}, it was shown that there exists a strongly regular Cayley graph with negative Latin square type parameters $(q^6,r(q^3+1),-q^3+r^2+3r,r^2+r)$, where $r=M(q^2-1)/2$, in the following cases: (i) $M=1$ and $q\equiv 3\,(\mod{4})$; (ii) $M=3$ and $q\equiv 7\,(\mod{24})$; and (iii) $M=7$ and $q\equiv 11,51\,(\mod{56})$. The existence of strongly regular Cayley graphs with the above parameters for odd $M>7$ was left open. In this paper, we prove that if there is an $h$, $1\le h\le M-1$, such that $M\,|\,(h^2+h+1)$ and the order of $2$ in $({\bf Z}/M{\bf Z})^\times$ is odd,then there exist infinitely many primes $q$ such that strongly regular Cayley graphs with the aforementioned parameters exist.