Triply Transitive Strongly-Regular Graphs

• Triply-transitive strongly-regular graphs are defined by the equality of the local Terwilliger, subalgebra, and point-stabilizer centralizer algebras, ensuring high local symmetry.
• The complete classification identifies eight families, including sporadic, product-type, and two infinite primitive geometric families from finite polar spaces.
• Structural methods combining Terwilliger algebra, orbital counting, and Krein parameters underpin the rigorous closure of the classification program.

Searching arXiv for the cited classification papers and the earlier program paper to ground the article. arxiv_search(query="triply-transitive strongly regular graphs Herman Maleki Razafimahatratra", max_results=10, sort_by="relevance") arxiv_search(query="The complete classification of triply-transitive strongly regular graphs", max_results=10, sort_by="relevance") Triply-transitive strongly-regular graphs are strongly regular graphs for which the local distance-partition algebra, the Terwilliger algebra, and the point-stabilizer centralizer algebra coincide at every base vertex. Concretely, if $\Gamma$ is vertex-transitive and for every $\omega\in V$ one has

$T_{0,\omega}=T_\omega=\widetilde T_\omega,$

then $\Gamma$ is triply-transitive. The condition isolates a highly constrained form of local symmetry, and its finite instances are now completely classified: Li and Zou proved the final open cases, completing a program initiated by Herman, Maleki, and Razafimahatratra (Li et al., 27 Oct 2025).

1. Formal definition and algebraic setting

A graph $\Gamma=(V,E)$ is strongly regular with parameters $(v,k,\lambda,\mu)$ if $|V|=v$, $\Gamma$ is $k$-regular, any two adjacent vertices have exactly $\lambda$ common neighbors, and any two non-adjacent vertices have exactly $\omega\in V$0 common neighbors. Equivalently, if $\omega\in V$1 is the adjacency matrix of $\omega\in V$2, then

$\omega\in V$3

Fix a vertex $\omega\in V$4, and write $\omega\in V$5 for $\omega\in V$6. The corresponding dual idempotents are the diagonal matrices

$\omega\in V$7

The Terwilliger algebra at $\omega\in V$8 is

$\omega\in V$9

and it contains the smaller subspace

$T_{0,\omega}=T_\omega=\widetilde T_\omega,$0

If $T_{0,\omega}=T_\omega=\widetilde T_\omega,$1 and $T_{0,\omega}=T_\omega=\widetilde T_\omega,$2 is the stabilizer of $T_{0,\omega}=T_\omega=\widetilde T_\omega,$3, then the centralizer algebra is

$T_{0,\omega}=T_\omega=\widetilde T_\omega,$4

One always has

$T_{0,\omega}=T_\omega=\widetilde T_\omega,$5

In this language, triply-transitive means vertex-transitive together with equality throughout the chain above. The earlier classification paper also records Munemasa’s criterion that $T_{0,\omega}=T_\omega=\widetilde T_\omega,$6 is triply-regular if and only if $T_{0,\omega}=T_\omega=\widetilde T_\omega,$7, so triply-transitivity adds the further requirement that the Terwilliger algebra coincide with the full point-stabilizer centralizer algebra (Herman et al., 18 Jul 2025).

2. Complete classification

The complete classification theorem states that a finite strongly regular graph is triply-transitive if and only if it belongs to one of eight families (Li et al., 27 Oct 2025).

Family	Description
(a)	Complete multipartite graph $T_{0,\omega}=T_\omega=\widetilde T_\omega,$8 with $T_{0,\omega}=T_\omega=\widetilde T_\omega,$9 parts
(b)	Pentagon $\Gamma$0
(c)	McLaughlin graph
(d)	Higman–Sims graph
(e)	Peisert graph $\Gamma$1
(f)	$\Gamma$2 grid graph
(g)	Collinearity graph of the polar space $\Gamma$3
(h)	Affine polar graph $\Gamma$4, $\Gamma$5, $\Gamma$6

This list is exhaustive: no other strongly regular graphs satisfy the triple-transitivity condition. Within the classification, the only infinite families of primitive triply-transitive strongly regular graphs are the geometric families $\Gamma$7 and $\Gamma$8. The remaining infinite examples are imprimitive product-type families, notably complete multipartite graphs and the $\Gamma$9 grids.

The earlier partial classification gives additional identifications for some entries. In particular, the $\Gamma=(V,E)$0 grid is the Hamming graph $\Gamma=(V,E)$1, and $\Gamma=(V,E)$2 is isomorphic to $\Gamma=(V,E)$3 (Herman et al., 18 Jul 2025).

3. Development of the classification program

The classification was obtained in two stages. Herman, Maleki, and Razafimahatratra initiated the program, and the 2025 paper “On the classification of triply-transitive strongly-regular graphs” classified all triply-transitive strongly regular graphs except two geometric families: the collinearity graph of the polar space $\Gamma=(V,E)$4 and the affine polar graph $\Gamma=(V,E)$5 (Herman et al., 18 Jul 2025). In that work, the complete multipartite graphs, $\Gamma=(V,E)$6, the McLaughlin graph, the Higman–Sims graph, $\Gamma=(V,E)$7, and the $\Gamma=(V,E)$8 grids were all verified to be triply-transitive.

The status of the subject therefore changed within 2025. Before Li and Zou’s completion theorem, the classification was conditional on two unresolved infinite families. After their work, those final cases were shown to satisfy

$\Gamma=(V,E)$9

and the classification became definitive (Li et al., 27 Oct 2025).

This chronology matters because it clarifies an otherwise easy source of confusion in the literature: statements of “classification” in mid-2025 still carried explicit exclusions, whereas the late-2025 result removes those exclusions and closes the program.

4. The polar-space family $(v,k,\lambda,\mu)$0

One of the two primitive infinite families arises from finite polar geometry. Let $(v,k,\lambda,\mu)$1 carry a nondegenerate quadratic form of minus type

$(v,k,\lambda,\mu)$2

where $(v,k,\lambda,\mu)$3 is any irreducible binary quadratic over $(v,k,\lambda,\mu)$4. The points of the polar space $(v,k,\lambda,\mu)$5 are the $(v,k,\lambda,\mu)$6-spaces $(v,k,\lambda,\mu)$7 with $(v,k,\lambda,\mu)$8. Two points $(v,k,\lambda,\mu)$9 are collinear if and only if the associated bilinear form satisfies $|V|=v$0 (Li et al., 27 Oct 2025).

Its collinearity graph is strongly regular with parameters

$|V|=v$1

The proof of triply-transitivity proceeds by fixing three special points

$|V|=v$2

and analyzing the subconstituents $|V|=v$3. The graph is primitive and vertex-transitive under $|V|=v$4. General theory gives $|V|=v$5 whenever both $|V|=v$6 and its complement contain triangles. Li and Zou then compute the block decomposition of $|V|=v$7 via the orbits of

$|V|=v$8

on $|V|=v$9. Their orbit-counting argument, using Witt’s theorem and certain field automorphisms, yields

$\Gamma$0

so $\Gamma$1. Hence $\Gamma$2, establishing triple-transitivity.

The smallest case is $\Gamma$3, where

$\Gamma$4

giving the unique $\Gamma$5.

5. The affine polar family $\Gamma$6

The second primitive infinite family is affine rather than projective. Let $\Gamma$7 carry a nondegenerate quadratic form of type $\Gamma$8,

$\Gamma$9

where

$k$0

The affine polar graph on $k$1 is defined by

$k$2

(Li et al., 27 Oct 2025).

This graph is strongly regular with parameters

$k$3

$k$4

$k$5

$k$6

The automorphism group is

$k$7

under which the graph is primitive and vertex-transitive. The proof again compares $k$8 with $k$9. By the relevant lemma for primitive strongly regular graphs, $\lambda$0 has dimension $\lambda$1 whenever both $\lambda$2 and its complement contain triangles, with one exceptional case $\lambda$3 where $\lambda$4. A rank-$\lambda$5 analysis of the orthogonal and non-orthogonal subgraphs gives

$\lambda$6

where $\lambda$7 is the number of orbits of $\lambda$8 on $\lambda$9. A direct orbit analysis, based on constructing explicit isometries that fuse candidate orbits, shows $\omega\in V$00. Comparison of the resulting dimensions yields $\omega\in V$01, and hence triple-transitivity.

Small examples include

$\omega\in V$02

the Clebsch graph,

$\omega\in V$03

and

$\omega\in V$04

6. Structural methods and mathematical significance

The classification rests on a combination of Terwilliger-algebra methods, orbital counting, and finite-geometry input. In the earlier analysis, the key structural observation is that $\omega\in V$05, so its dimension is the number of orbitals of $\omega\in V$06 on $\omega\in V$07. In rank $\omega\in V$08, the corresponding block form is always

$\omega\in V$09

and equality of dimensions forces

$\omega\in V$10

(Herman et al., 18 Jul 2025).

The same paper also uses dimension counts for the Peirce blocks $\omega\in V$11, where $\omega\in V$12, together with Krein-parameter arguments. In primitive cases, one obtains $\omega\in V$13, while in imprimitive cases $\omega\in V$14. The Krein-parameter test sharply restricts which primitive rank-$\omega\in V$15 families can be triply-regular, and the subsequent comparison with $\omega\in V$16 isolates the triply-transitive cases.

The completed classification therefore exhibits a precise boundary phenomenon. Triply-transitivity is much more restrictive than strong regularity alone and remains rare even among highly symmetric graphs. The final list combines sporadic examples, product-type families, and two genuinely geometric primitive infinite families. This suggests a close alignment between local algebraic symmetry, rank-$\omega\in V$17 permutation structure, and the incidence geometry of polar spaces, an alignment made explicit by the equality $\omega\in V$18 (Li et al., 27 Oct 2025).