Van Lint–Schrijver Graphs

• Van Lint–Schrijver graphs are highly symmetric Cayley graphs defined over finite fields using a unique subgroup structure.
• They exhibit strong regularity and rank-three properties, linking finite geometry, algebraic graph theory, and combinatorial designs.
• These graphs serve as benchmarks for studying automorphism groups, critical groups, and algorithmic Hamilton cycle constructions.

Van Lint–Schrijver graphs, originating from the combinatorial and algebraic studies of J.H. van Lint and A. Schrijver, constitute a pivotal family within the context of rank-three finite affine Cayley graphs and highly symmetric strongly regular graphs. These graphs unify several deep connections among finite geometry, algebraic graph theory, combinatorics, and coding theory. They also provide critical examples in the paper of colorability, automorphism group structure, critical groups, and extremal combinatorics.

1. Explicit Construction, Definitions, and Classification

Van Lint–Schrijver graphs are most precisely defined as follows. Given primes $p$ and $\ell>2$ with $p$ primitive modulo $\ell$, and an integer $t\ge 1$, set $q=p^{(\ell-1)t}$ and let $K=\mathbb{F}_q$. Let $S\subset K^\times$ be the unique multiplicative subgroup of order $k=(q-1)/\ell$. Then the Van Lint–Schrijver graph is the undirected Cayley graph $G(p,\ell,t)=\mathrm{Cay}(K,S)$ with vertex set $K$ and edges $\{x,y\}$ if and only if $x-y\in S$ (Muzychuk, 2020).

These graphs are a subclass of generalized Paley graphs $G(q,k)=\mathrm{Cay}(F_q, S)$, where $S=\{z^k: z\in F_q^*\}$ for prime power $q$ and divisor $k$ of $q-1$ (Ponomarenko, 23 Nov 2025). The automorphism group is, for large $q$, a subgroup of the affine semilinear group $A\Gamma L(1,q)$ (Ponomarenko, 23 Nov 2025).

Muzychuk (Muzychuk, 2020) established that every one-dimensional affine rank-three graph on $\mathbb{F}_q$ is either a Paley graph ($\ell=2$), a Van Lint–Schrijver graph ($\ell>2$), or a Peisert graph (the index-4 special case), up to complementation, yielding a complete classification in the rank-three affine context.

2. Strongly Regular Parameters and Spectral Structure

Van Lint–Schrijver graphs are strongly regular with parameters $(v,k,\lambda,\mu)$: $v=q,\quad k=(q-1)/\ell$

$$\lambda=\frac{q-3\ell+1+(\ell-2)(\ell-1)\sqrt{q}}{\ell^2},\quad \mu=\frac{q-\ell+1-(\ell-2)\sqrt{q}}{\ell^2}$$

Their adjacency matrix has exactly three eigenvalues: $k$, $r_1=\frac{-1+(\ell-1)\sqrt{q}}{\ell}$, $r_2=\frac{-1-\sqrt{q}}{\ell}$, with multiplicities $1$, $(q-1)/\ell$, and $(\ell-1)(q-1)/\ell$ respectively (Muzychuk, 2020, Pantangi, 2018). The explicit spectral formulas derive from character theory on finite fields, including Gauss sum evaluations.

3. Automorphism Group, Rank-three Property, and WL Complexity

The automorphism group of a Van Lint–Schrijver graph is, generically, the affine semilinear group $$\mathrm{A}\Gamma \mathrm{L}(1,q)=F_q^+ \ltimes \Gamma L(1,q)$$, where $\Gamma L(1,q)=F_q^* \ltimes \mathrm{Gal}(F_q/F_p)$ acts by field automorphisms and scaling (Ponomarenko, 23 Nov 2025). For sufficiently large $q$ relative to $k$, this automorphism group contains all possible symmetries (Ponomarenko, 23 Nov 2025, Muzychuk, 2020).

In Weisfeiler–Leman (WL) graph isomorphism complexity, the graphs have WL-dimension at most $5$ and at least $3$ for large $q$; thus, they are not distinguished by 2-dimensional WL but are by higher-dimensional refinements (Ponomarenko, 23 Nov 2025). This situates Van Lint–Schrijver graphs with Paley and Peisert graphs as the canonical rank-3 affine Cayley graphs.

4. Partial Geometry: Van Lint–Schrijver (5,5,2) and the srg(81,30,9,12)

The classical srg(81, 30, 9, 12) introduced by van Lint and Schrijver arises as the Cayley graph on $\mathbb{F}_3^4$ with connection set derived from a ternary [30,4,9] two-weight code (Crnković et al., 2020). These graphs are geometric, possessing a partial geometry $\mathrm{pg}(5,5,2)$ with 81 points and lines, each line incident with six points, each point on six lines, and with exactly two bridging lines between any point not on a given line.

Explicitly, the parameters are: $$v = 81,\quad k = 30,\quad \lambda = 9,\quad \mu = 12$$ The automorphism group for this partial geometry is flag-transitive and of order $58,320$ (Crnković et al., 2020, Winter et al., 2020). The line-stabilizer is trivial, corresponding to the "rigid type" Singer abelian group property. Uniqueness up to $10^6$ points for partial geometries of rigid type is conjectured, with only three hypothetical non-unique examples beyond Van Lint–Schrijver known (Winter et al., 2020). The critical case (5,5,2) is realized in GF(3)^4.

5. Critical Group and Smith Normal Form

The critical group $\operatorname{Crit}(G)$ for $G=G(p,\ell,t)$ is determined by the Smith normal form of the Laplacian matrix. The spectrum involves the block-diagonalization under the action of $S$ (multiplicative subgroup), and Smith invariants are expressed in terms of $p$-adic properties of Jacobi sums and Stickelberger's theorem (Pantangi, 2018). The group decomposes as a direct sum of Sylow $p$-part and prime-to-$p$ part, with explicit formulas for elementary divisor multiplicities and link to the number of spanning trees.

6. Connections to Schrijver Graphs, Vertex- and Edge-criticality

Schrijver graphs $SG(n,k)$, also called stable Kneser graphs, are induced subgraphs of the classical Kneser graph on $k$-subsets of $[n]$, restricting to subsets avoiding any two cyclically consecutive elements. They realize chromatic number $n-2k+2$, as established via topological methods by Lovász and refined for vertex-criticality by Schrijver (Kaiser et al., 2019, Kaiser et al., 2020). For $k=2$ and general $k$, sharp constructions of edge-critical subgraphs are known, with explicit combinatorial (alternator and necklace factor) criteria, and relate to Van Lint–Schrijver graphs via their extremal colorability and stability (Kaiser et al., 2019, Kaiser et al., 2020, Mütze et al., 3 Jan 2024).

7. Hamiltonicity and Combinatorial Generation

Every Schrijver graph $S(n,k)=S(n,k,2)$ and its generalizations ($s$-stable Kneser graphs) admit an explicit Hamilton cycle, found via combinatorial constructions involving necklace (orbit) decompositions and cycle gluing via connector 4-cycles (Mütze et al., 3 Jan 2024). An associated shift-rule algorithm generates the Hamilton cycle with $O(n)$ time per vertex, generalizing Gray-code and necklace enumeration for arbitrary stability parameters, highlighting the algorithmic efficiency encoded in the structural symmetries of the graphs.

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The Van Lint–Schrijver graphs occupy a central role at the intersection of algebraic graph theory (rank-three Cayley graphs, srg classification), finite geometry (partial geometries, rigid-type Singer actions), combinatorics (Kneser-type stability, chromatic and Hamiltonicity extremality), and algebraic invariants (critical group/SNF analysis). Their properties—strong regularity, high symmetry, extremal colorability, rigid group actions, Hamiltonicity—render them archetypal test cases and benchmarks for theoretical and computational advances across several domains of combinatorics and discrete mathematics.