Tangent Graphs in Finite Classical Polar Spaces

• Tangent graphs are combinatorial structures defined on the external points of a finite classical polar space, with edges determined by line tangencies.
• They exhibit strongly regular graph properties with explicit spectra and parameters, linking finite geometry to Ramanujan graph theory.
• Multiple equivalent constructions—geometric, algebraic, and incidence-based—generalize these graphs to infinite families with applications in coding theory and combinatorial designs.

A tangent graph of a finite classical polar space is a combinatorial structure arising from finite projective geometries equipped with a nondegenerate form, and encodes adjacencies between external points based on tangency of lines to the underlying polar space. Recent developments have illuminated the spectral and expansion properties of these graphs, particularly their connection to Ramanujan graphs and strongly regular graphs (SRGs), with explicit constructions and parameterizations for orthogonal and unitary types over small fields such as $\operatorname{GF}(2)$ and $\operatorname{GF}(4)$ (Smaldore, 17 Jan 2026, Smaldore, 20 Sep 2025).

1. Classical Polar Spaces and Ambient Geometry

Finite classical polar spaces are sets of subspaces in projective geometry invariant under a suitable nondegenerate form. The principal families relevant to tangent graph constructions are:

• Symplectic space $W(2m-1,q)$: defined by totally isotropic subspaces with respect to an alternating bilinear form in $\operatorname{PG}(2m-1,q)$.
• Orthogonal quadrics $Q^\varepsilon$ in $\operatorname{PG}(N,q)$: defined by totally singular subspaces with respect to a quadratic form; $\varepsilon = +$ or $-$ denotes the Witt index for $N$ odd, with the parabolic case for even $N$.
• Hermitian (unitary) spaces $H(N,q^2)$: consisting of totally isotropic subspaces under a nondegenerate Hermitian form.

Canonical names are used for small-field cases: $Q^\pm(2m-1,2)$ for orthogonal, $Q(2m,2)$ for parabolic, and $H(m-1,4)$ for unitary types (Smaldore, 17 Jan 2026).

2. Definition, Construction, and Properties of Tangent Graphs

Let $X$ be a finite classical polar space embedded in a projective space over the finite field $\mathbb{F}_q$. The tangent graph $\operatorname{Tang}(X)$ is defined as follows:

• Vertices: the set $V = \operatorname{PG} \setminus X$ of all non-isotropic points.
• Edges: distinct $P, Q \in V$ are adjacent if and only if the projective line $\langle P, Q \rangle$ is tangent to $X$, i.e., meets $X$ in exactly one point.

For all the families treated, the tangent graph $\operatorname{Tang}(X)$ is a strongly regular graph with explicitly computable parameters (order $v$, degree $k$, eigenvalues $\lambda_2, \lambda_3$ and multiplicities $m_2, m_3$). The full spectrum in closed form is provided for orthogonal (even/odd-dimension and both types), and unitary cases (Smaldore, 17 Jan 2026).

Example: The Tangent Graph $NO^+(6,2)$

Let $V = \mathbb{F}_2^6$ with the hyperbolic quadric $Q^+(5,2) \subset \operatorname{PG}(5,2)$. The tangent graph $NO^+(6,2)$ is constructed on $V = \operatorname{PG}(5,2) \setminus Q^+(5,2)$, with edges induced by tangency as above. Parameters are:

• $v = 28$
• $k = 15$
• $\lambda = 6$
• $\mu = 10$
• Spectrum: $(15)^1, (1)^{20}, (-5)^7$

The automorphism group is $PGO^+(6,2) \cong O^+(6,2) \cong A_8$ (Smaldore, 20 Sep 2025).

3. Equivalent Constructions and Models

Multiple, equivalently isomorphic constructions for the same tangent graph arise from different geometric, combinatorial, and algebraic frameworks. For $NO^+(6,2)$, six distinct constructions are catalogued (Smaldore, 20 Sep 2025):

• Incidence-geometric ("quadric with a hole" and complement): Remove a generator plane from $Q^+(5,2)$, consider points outside, and define adjacency via inclusion of lines in the quadric or secancy with intersection in the generator.
• Klein correspondence (projective-line model): Lines of $\operatorname{PG}(3,2)$ mapped into $Q^+(5,2)$ under the Plücker embedding; adjacency induced by line intersections or skewness.
• Point–line antiflags in $\operatorname{PG}(2,2)$: The set of $28$ antiflags forms an association scheme whose union of three of the four relations yields the graph.
• Secant variety of Veronese surface in $\operatorname{PG}(5,2)$: Points of $\operatorname{PG}(5,2)\setminus$ the secant variety, with edges determined by line interaction.
• Non-singular $3 \times 3$ symmetric matrices: Matrices of full rank over $\mathbb{F}_2$, adjacent when their sum is singular.

All constructions ultimately produce the same SRG, and generalize naturally as $n$ and $q$ vary, for example, via the antiflag association for general $NO^+(2n,2)$ or matrix model for higher sizes (Smaldore, 20 Sep 2025).

4. Parameters, Spectra, and Automorphism Groups

Explicit closed formulas are provided for the parameter sets governing tangent graphs of classical polar spaces (Smaldore, 17 Jan 2026):

• For $NO^\varepsilon(2m,2)$:
  • $v = 2^{2m-1} - \varepsilon 2^{m-1}$
  • $k = 2^{2m-2} - 1$
  • $\lambda_2 = \varepsilon 2^{m-2} - 1$
  • $\lambda_3 = -\varepsilon 2^{m-1} - 1$
  • Spectrum: $$(k)^1, (\varepsilon 2^{m-2} - 1)^{m_2}, (-\varepsilon 2^{m-1} - 1)^{m_3}$$
• For $NO(2m+1,2)$ (parabolic case): analogous parameterization and spectral data.
• For $NU(m,4)$ (unitary):
  • $v = 2^{2m-1} - \varepsilon 2^{m-1}$
  • $k = (2^{m-1} + \varepsilon)(2^{m-2} - \varepsilon)$
  • Eigenvalues: $\lambda_2 = \varepsilon 2^{m-2} - 1$, $\lambda_3 = -\varepsilon 2^{m-3} - 1$

Automorphism groups are described precisely; for $NO^+(6,2)$, the automorphism group is $A_8$ (Smaldore, 20 Sep 2025).

5. Ramanujan Property and Expansion

A $d$-regular graph $G$ is Ramanujan if every eigenvalue $\lambda_i \ne \pm d$ of its adjacency matrix satisfies $|\lambda_i| \le 2\sqrt{d-1}$. For tangent graphs of classical polar spaces, this criterion is verified explicitly:

• For $NO^\varepsilon(2m,2)$, $NO(2m+1,2)$, $NU(m,4)$, all (except for certain small values of $m$) satisfy the Ramanujan bound, as direct algebraic checks of the squared inequalities confirm (Smaldore, 17 Jan 2026).

Family	Range of $m$	Ramanujan?
$NO^+(2m,2)$	$m=1$: trivial, $m=2$: $K_{3,3}$	$m\ge 3$: yes
$NO^-(2m,2)$	$m=1$: $3K_1$, $m=2$: Petersen	$m\ge 2$: yes
$NO(2m+1,2)$	$m=1$: $3K_1$	$m\ge 2$: yes
$NU(m,4)$	$m=2$: $2K_1$, $m=3$: $\overline{4K_3}$	$m\ge 3$: yes

Satisfaction of the Ramanujan property implies optimal expansion, as the non-trivial eigenvalues achieve the bound imposed by the theory of expanders. In the binary and unitary cases, these infinite families provide new explicit constructions of Ramanujan SRGs (Smaldore, 17 Jan 2026).

6. Generalizations and Further Directions

All described models—incidence-geometric, Klein, antiflag, Veronese, and matrix—naturally generalize to higher rank and field size. The parameter schemes and adjacency criteria extend to yield infinite families of strongly regular tangent graphs for $NO^+(2n,q)$ and related types. For $q=2$ and higher $n$, these generalizations continue to produce Ramanujan graphs, with explicit spectrum and automorphism groups. A plausible implication is that these constructions provide fertile ground for applications to coding theory and combinatorial designs, given the close structural links to expander properties and association schemes (Smaldore, 17 Jan 2026, Smaldore, 20 Sep 2025).

7. Summary of Key Results and Proof Strategies

All tangent graphs of the discussed types are shown to be SRGs with spectra computable in closed form. The eigenvalue bounds needed for the Ramanujan property reduce to comparing powers of $2$ or $4$, enabling straightforward algebraic verification. Small cases are handled by direct computation or correspondence to known graph types (complete, bipartite, or Petersen graphs). The classification is exhaustive for the orthogonal (binary) and unitary ($q=2$) families. The underlying rationale leverages the tight connection between polar space combinatorics and spectral graph theory (Smaldore, 17 Jan 2026).

Collectively, these results establish the tangent graphs of finite classical polar spaces—when properly parameterized and excepting several small cases—as infinite families of Ramanujan strongly regular graphs, with explicit algebraic, geometric, and combinatorial realizations that generalize to broad classes of parameters (Smaldore, 17 Jan 2026, Smaldore, 20 Sep 2025).