[Paper Review] Constructions of Strongly Regular Cayley Graphs using Even Index Gauss Sums

Fan Wu|arXiv (Cornell University)|Aug 5, 2012

Coding theory and cryptography12 references3 citations

TL;DR

This paper generalizes constructions of strongly regular Cayley graphs using even-index Gauss sums, providing necessary and sufficient conditions for such graphs to exist. It introduces three new infinite families of strongly regular graphs by leveraging cyclotomic classes and subfield properties, extending prior work on index 2 and 4 cases to higher even indices, including a novel generalization of an index 6 example not covered by earlier methods.

ABSTRACT

In this paper, generalizing the result in \cite{GXY}, we construct strongly regular Cayley graphs by using union of cyclotomic classes of $\F_q$ and Gauss sums of index $w$, where $w\geq 2$ is even. In particular, we obtain three infinite families of strongly regular graphs with new parameters.

Motivation & Objective

• To extend the construction of strongly regular Cayley graphs beyond index 2 and 4 using Gauss sums of higher even indices.
• To establish necessary and sufficient conditions for a union of cyclotomic classes to yield a strongly regular Cayley graph.
• To generalize sporadic examples from the Schmidt-White conjecture into infinite families, particularly for index 6 and higher.
• To reveal a structural link between strongly regular Cayley graphs and cyclic difference sets in (Z/p₁Z, +).
• To provide a direct construction method that avoids explicit evaluation of high-index Gauss sums, instead relying on subfield membership.

Proposed method

• Generalizing prior work, the paper constructs Cayley graphs over F_q using unions of cyclotomic classes of index w ≥ 2 even.
• It introduces a key condition (Theorem 3.3) that determines when such unions yield strongly regular graphs based on Gauss sum properties.
• The method relies on analyzing Gauss sums' subfield membership rather than explicit evaluation, simplifying the construction for high indices.
• It uses the canonical additive character ψ and multiplicative characters of F_q to define Gauss sums and derive spectral conditions.
• The construction leverages the structure of multiplicative subgroups D of F_q* of index w, with D = -D, and relates them to subfields via the index w and prime power p₁.
• Corollary 3.5 establishes a reduction: if the base case over F_{p^{f̃}} yields an srg, then so does the lifted construction over F_q.

Experimental results

Research questions

• RQ1 Can constructions of strongly regular Cayley graphs using Gauss sums be generalized from index 2 and 4 to higher even indices, such as 6 or more?
• RQ2 What are the necessary and sufficient conditions for a union of cyclotomic classes of even index w to generate a strongly regular Cayley graph?
• RQ3 How can the evaluation of high-index Gauss sums be avoided in such constructions while still ensuring strong regularity?
• RQ4 Is there a structural connection between the resulting strongly regular graphs and cyclic difference sets in (Z/p₁Z, +)?
• RQ5 Can sporadic examples from the Schmidt-White conjecture, particularly index 6 cases, be generalized into infinite families using this method?

Key findings

• The paper constructs three new infinite families of strongly regular Cayley graphs using even-index Gauss sums, generalizing known sporadic examples.
• Example 4.1 provides a new infinite family that generalizes Example 5 from Table I in the Schmidt-White conjecture, with parameters derived from p=11, p₁=43, w=6.
• Example 4.2 generalizes subfield examples by constructing srgs over F_{5^f} with w=10, where the base graph over F_{5^3} is the subfield srg with D = F_5*.
• Example 4.3 constructs srgs over F_{2^f} with w=18, where the base graph over F_{2^7} is trivially srg since D = {1}, the multiplicative group of the prime field.
• Theorem 3.3 provides a complete characterization: a Cayley graph is strongly regular if and only if specific conditions on Gauss sums and subfield embeddings are satisfied.
• The results reveal a deep connection between strongly regular Cayley graphs and cyclic difference sets in (Z/p₁Z, +), as encoded in conditions (3.16) and (3.17).