[Paper Review] Strongly regular Cayley graphs from partitions of subdifference sets of the Singer difference sets

Koji Momihara, Qing Xiang|arXiv (Cornell University)|Jun 17, 2017

Coding theory and cryptography6 references3 citations

TL;DR

This paper presents a new lifting construction of hyperbolic-type strongly regular Cayley graphs over finite fields using partitions of subdifference sets derived from Singer difference sets. By generalizing previous constructions related to $m$-ovoids and $i$-tight sets, the authors unify and simplify existing results, yielding new or nonisomorphic strongly regular graphs with known parameters, including examples over $\mathbb{F}_{q^{2m}}$ with parameters matching affine polar graphs.

ABSTRACT

In this paper, we give a new lifting construction of "hyperbolic" type of strongly regular Cayley graphs. Also we give new constructions of strongly regular Cayley graphs over the additive groups of finite fields based on partitions of subdifference sets of the Singer difference sets. Our results unify some recent constructions of strongly regular Cayley graphs related to $m$-ovoids and $i$-tight sets in finite geometry. Furthermore, some of the strongly regular Cayley graphs obtained in this paper are new or nonisomorphic to known strongly regular graphs with the same parameters.

Motivation & Objective

• To develop a new lifting construction for strongly regular Cayley graphs of hyperbolic type.
• To unify recent constructions of strongly regular Cayley graphs related to $m$-ovoids and $i$-tight sets in finite geometry.
• To provide simpler proofs for existing constructions using subdifference sets of Singer difference sets.
• To generate new or nonisomorphic strongly regular graphs with parameters matching known families like affine polar graphs.

Proposed method

• Lift cyclotomic strongly regular graphs from $\mathbb{F}_q$ to $\mathbb{F}_{q^m}$ using a generalized partition of subdifference sets of Singer difference sets.
• Apply Gauss sum techniques and character sum analysis over finite fields to verify eigenvalue conditions for strong regularity.
• Use the structure of subfields and trace maps to partition subdifference sets and control character sums.
• Leverage the properties of cyclotomic classes $C_t^{(N,q^m)}$ and their unions to define connection sets for Cayley graphs.
• Verify that the resulting Cayley graphs satisfy the necessary eigenvalue and adjacency matrix conditions for strong regularity.
• Prove that the character sums of the connection set $D$ yield only two distinct restricted eigenvalues, confirming strong regularity.

Experimental results

Research questions

• RQ1 Can a new lifting construction of hyperbolic-type strongly regular Cayley graphs be developed from subdifference sets of Singer difference sets?
• RQ2 How can constructions based on $m$-ovoids and $i$-tight sets be unified and simplified using subdifference set partitions?
• RQ3 Do the new constructions yield strongly regular graphs that are nonisomorphic to known graphs with the same parameters?
• RQ4 What role do Gauss sums and trace maps play in characterizing the eigenvalues of these Cayley graphs?
• RQ5 Under what conditions on $q$ and $m$ do the resulting graphs achieve Latin square or negative Latin square type parameters?

Key findings

• A new hyperbolic-type lifting construction of strongly regular Cayley graphs is established, extending prior elliptic-type constructions.
• For $q \equiv 3 \pmod{4}$ and odd $m > 1$, a $(q^{2m}, r(q^m+1), -q^m + r^2 + 3r, r^2 + r)$ strongly regular Cayley graph exists with $r = q^{m-1}(q-1)/2$.
• For $q \equiv 1 \pmod{4}$ and odd $m > 1$ with $\gcd(q-1, (q^m-1)/(q-1)) = 1$, a $(q^{2m}, r(q^m-1), q^m + r^2 - 3r, r^2 - r)$ strongly regular Cayley graph exists with $r = q^{m-1}(q-1)/2$.
• The graph constructed for $q=3$, $m=3$ is not isomorphic to the affine polar graph ${\mathrm{AP}}^{-}$ with the same parameters, as confirmed by automorphism group size comparison.
• The graph constructed for $q=5$, $m=3$ is not isomorphic to the affine polar graph ${\mathrm{AP}}^{+}$ with the same parameters, again confirmed by distinct automorphism group orders.
• The results unify and simplify recent constructions related to $m$-ovoids and $i$-tight sets by generalizing them through subdifference set partitions of Singer difference sets.