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Switching equivalence of strongly regular polar graphs

Gábor P. Nagy, Valentino Smaldore

TL;DR

The paper establishes the switching equivalence of the strongly regular polar graphs $NO^{\pm}(4m,2)$ and $NO^{\mp}(2m+1,4)$ by deriving analytic and two-graph descriptions that connect Seidel switching to polar-geometry data in characteristic two. It develops the symplectic two-graph framework, augments it with descendants, and proves that the two graphs share the same associated two-graph, leading to a concrete Seidel switching set described via a quadratic form over $\mathbb{F}_4$. The result yields explicit parameters for the related switched graphs and deepens the understanding of how polar geometries encode cospectral, switching-related equivalences. The methods combine detailed group-action analysis with quadrics in characteristic two to produce a precise, computable equivalence that has implications for the broader study of cospectrality and graph isomorphism within polar spaces.

Abstract

We prove the switching equivalence of the strongly regular polar graphs $NO^\pm(4m,2)$, $NO^\mp(2m+1,4)$, and $Γ(O^\mp(4m,2))$ plus an isolated vertex by giving an analytic description for them and their associated two-graphs.

Switching equivalence of strongly regular polar graphs

TL;DR

The paper establishes the switching equivalence of the strongly regular polar graphs and by deriving analytic and two-graph descriptions that connect Seidel switching to polar-geometry data in characteristic two. It develops the symplectic two-graph framework, augments it with descendants, and proves that the two graphs share the same associated two-graph, leading to a concrete Seidel switching set described via a quadratic form over . The result yields explicit parameters for the related switched graphs and deepens the understanding of how polar geometries encode cospectral, switching-related equivalences. The methods combine detailed group-action analysis with quadrics in characteristic two to produce a precise, computable equivalence that has implications for the broader study of cospectrality and graph isomorphism within polar spaces.

Abstract

We prove the switching equivalence of the strongly regular polar graphs , , and plus an isolated vertex by giving an analytic description for them and their associated two-graphs.
Paper Structure (16 sections, 24 theorems, 65 equations)

This paper contains 16 sections, 24 theorems, 65 equations.

Table of Contents

  1. Symplectic and orthogonal groups in characteristic two
  2. Quadratic forms and alternating bilinear forms
  3. Symplectic and orthogonal transformations
  4. Strongly regular graphs, Seidel switching, two-graphs
  5. Preliminaries on projective quadrics
  6. Strongly regular polar graphs
  7. Two-graphs and switchings
  8. Analytic description of strongly regular polar graphs
  9. Two-graphs of strongly regular polar graphs
  10. The symplectic two-graphs
  11. The descendants of symplectic two-graphs
  12. The main result on the switching equivalence
  13. Acknowledgement
  14. Symplectic groups in even characteristic
  15. Symplectic transvections
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $q$ be a power of two, and $\langle .,. \rangle$ a symplectic bilinear form on $\mathbb{F}_q^{2m}$. Let $\Omega$ be the set of quadratic forms $V\to \mathbb{F}_q$ which linearize to $\langle .,. \rangle$.

Theorems & Definitions (49)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Definition 2.1
  • Lemma 2.2
  • Definition 2.3
  • Lemma 2.4
  • Definition 2.5: Two-graph
  • Definition 2.6: Associated two-graph
  • Definition 2.7: Seidel switching seidel
  • ...and 39 more