---
title: Triply Transitive Strongly-Regular Graphs
url: https://www.emergentmind.com/topics/triply-transitive-strongly-regular-graphs
type: topic
---

# Triply Transitive Strongly-Regular Graphs

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 [2510.23441].

## 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 \(\mu\) common neighbors. Equivalently, if \(A\) is the adjacency matrix of \(\Gamma\), then
\[
A^2 = kI + \lambda A + \mu(J-I-A).
\]

Fix a vertex \(\omega\in V\), and write \(\Delta_i=\{x:\operatorname{dist}(\omega,x)=i\}\) for \(i=0,1,2\). The corresponding dual idempotents are the diagonal matrices
\[
E_i^*=\operatorname{diag}(1_{\operatorname{dist}(\omega,x)=i}:x\in V).
\]
The Terwilliger algebra at \(\omega\) is
\[
T_\omega=\langle A,E_0^*,E_1^*,E_2^*\rangle\subseteq \operatorname{Mat}_v(\mathbb C),
\]
and it contains the smaller subspace
\[
T_{0,\omega}=\operatorname{span}\{E_i^*A^jE_k^*: i,j,k\in\{0,1,2\}\}.
\]

If \(G=\operatorname{Aut}(\Gamma)\) and \(G_\omega\) is the stabilizer of \(\omega\), then the centralizer algebra is
\[
\widetilde T_\omega=\operatorname{End}_{G_\omega}(\mathbb C^V)
=\{M\in \operatorname{Mat}_v(\mathbb C):P_gM=MP_g\ \forall g\in G_\omega\}.
\]
One always has
\[
T_{0,\omega}\subseteq T_\omega\subseteq \widetilde T_\omega.
\]

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 \(\Gamma\) is triply-regular if and only if \(T_{0,\omega}=T_\omega\), so triply-transitivity adds the further requirement that the Terwilliger algebra coincide with the full point-stabilizer centralizer algebra [2507.14320].

## 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 [2510.23441].

| Family | Description |
|---|---|
| (a) | Complete multipartite graph \(K_{m,\ldots,m}\) with \(n\) parts |
| (b) | Pentagon \(C_5\) |
| (c) | McLaughlin graph |
| (d) | Higman–Sims graph |
| (e) | Peisert graph \(P^*(9)\) |
| (f) | \(n\times n\) grid graph |
| (g) | Collinearity graph of the polar space \(\mathcal Q^{-}(5,q)\) |
| (h) | Affine polar graph \(\mathrm{VO}^{\varepsilon}_{2m}(2)\), \(m\ge 2\), \(\varepsilon=\pm 1\) |

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 \((g)\) and \((h)\). The remaining infinite examples are imprimitive product-type families, notably complete multipartite graphs and the \(n\times n\) grids.

The earlier partial classification gives additional identifications for some entries. In particular, the \(n\times n\) grid is the Hamming graph \(H(2,n)\), and \(P^*(9)\) is isomorphic to \(\operatorname{Paley}(9)\) [2507.14320].

## 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 \(O_6^-(q)\) and the affine polar graph \(\mathrm{VO}_{2m}^{\varepsilon}(2)\) [2507.14320]. In that work, the complete multipartite graphs, \(C_5\), the McLaughlin graph, the Higman–Sims graph, \(P^*(9)\), and the \(n\times n\) 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
\[
T_{0,\omega}=T_\omega=\widetilde T_\omega,
\]
and the classification became definitive [2510.23441].

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 \(\mathcal Q^{-}(5,q)\)

One of the two primitive infinite families arises from finite polar geometry. Let \(V=\mathbb F_q^6\) carry a nondegenerate quadratic form of minus type
\[
Q(x)=x_1x_2+x_3x_4+f(x_5,x_6),
\]
where \(f\) is any irreducible binary quadratic over \(\mathbb F_q\). The points of the polar space \(\mathcal Q^{-}(5,q)\) are the \(1\)-spaces \(\langle x\rangle\subseteq V\) with \(Q(x)=0\). Two points \(\langle x\rangle,\langle y\rangle\) are collinear if and only if the associated bilinear form satisfies \(B(x,y)=0\) [2510.23441].

Its collinearity graph is strongly regular with parameters
\[
v=(q+1)(q^3+1),\qquad
k=q(q^2+1),\qquad
\lambda=q-1,\qquad
\mu=q^2+1.
\]

The proof of triply-transitivity proceeds by fixing three special points
\[
\omega=\langle e_1\rangle,\qquad
\omega_1=\langle e_3\rangle,\qquad
\omega_2=\langle e_2\rangle,
\]
and analyzing the subconstituents \(\Delta_i=\{x:\operatorname{dist}(\omega,x)=i\}\). The graph is primitive and vertex-transitive under \(G=P\Gamma O_6^-(q)\). General theory gives \(\dim T_{0,\omega}=15\) whenever both \(\Gamma\) and its complement contain triangles. Li and Zou then compute the block decomposition of \(\widetilde T_\omega\) via the orbits of
\[
H_2=G_\omega\cap G_{\omega_2}
\qquad\text{and}\qquad
H_1=G_\omega\cap G_{\omega_1}
\]
on \(\Delta_1,\Delta_2\). Their orbit-counting argument, using Witt’s theorem and certain field automorphisms, yields
\[
D(\widetilde T_\omega)=
\begin{bmatrix}
1&1&1\\
1&3&2\\
1&2&3
\end{bmatrix},
\]
so \(\dim \widetilde T_\omega=15\). Hence \(T_{0,\omega}=\widetilde T_\omega\), establishing triple-transitivity.

The smallest case is \(q=2\), where
\[
v=27,\qquad k=10,\qquad \lambda=1,\qquad \mu=5,
\]
giving the unique \(\operatorname{srg}(27,10,1,5)\).

## 5. The affine polar family \(\mathrm{VO}^{\varepsilon}_{2m}(2)\)

The second primitive infinite family is affine rather than projective. Let \(V=\mathbb F_2^{2m}\) carry a nondegenerate quadratic form of type \(\varepsilon=\pm 1\),
\[
Q(x)=x_1x_2+\cdots +x_{2m-3}x_{2m-2}+f(x_{2m-1},x_{2m}),
\]
where
\[
f(x,y)=xy \quad\text{if }\varepsilon=+1,
\qquad\text{or}\qquad
f(x,y)=x^2+xy+y^2 \quad\text{if }\varepsilon=-1.
\]
The affine polar graph on \(V\) is defined by
\[
u\sim v \iff Q(u-v)=0
\]
[2510.23441].

This graph is strongly regular with parameters
\[
v=2^{2m},
\]
\[
k=(2^m-\varepsilon)(2^{m-1}+\varepsilon),
\]
\[
\lambda=2(2^{m-1}-\varepsilon)(2^{m-2}+\varepsilon),
\]
\[
\mu=2^{m-1}(2^{m-1}+\varepsilon).
\]

The automorphism group is
\[
G=T_V\rtimes GO_{2m}^{\varepsilon}(2),
\]
under which the graph is primitive and vertex-transitive. The proof again compares \(\dim T_{0,\omega}\) with \(\dim \widetilde T_\omega\). By the relevant lemma for primitive strongly regular graphs, \(T_{0,\omega}\) has dimension \(15\) whenever both \(\Gamma\) and its complement contain triangles, with one exceptional case \(m=2,\varepsilon=-1\) where \(\dim T_{0,\omega}=14\). A rank-\(3\) analysis of the orthogonal and non-orthogonal subgraphs gives
\[
D(\widetilde T_\omega)=
\begin{bmatrix}
1&1&1\\
1&3&t\\
1&t&3
\end{bmatrix},
\]
where \(t\) is the number of orbits of \(H_2\) on \(\Delta_1\). A direct orbit analysis, based on constructing explicit isometries that fuse candidate orbits, shows \(t=2\). Comparison of the resulting dimensions yields \(T_{0,\omega}=\widetilde T_\omega\), and hence triple-transitivity.

Small examples include
\[
m=2,\ \varepsilon=-1:\ \operatorname{srg}(16,5,0,2),
\]
the Clebsch graph,
\[
m=2,\ \varepsilon=+1:\ \operatorname{srg}(16,6,2,2),
\]
and
\[
m=3,\ \varepsilon=+1:\ \operatorname{srg}(64,28,12,12).
\]

## 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 \(\widetilde T_\omega=\operatorname{End}_{G_\omega}(\mathbb C^\Omega)\), so its dimension is the number of orbitals of \(G_\omega\) on \(\Omega\). In rank \(3\), the corresponding block form is always
\[
\begin{bmatrix}
1&1&1\\
1&r_1&t\\
1&t&r_2
\end{bmatrix},
\]
and equality of dimensions forces
\[
T_{0,\omega}=T_\omega=\widetilde T_\omega
\]
[2507.14320].

The same paper also uses dimension counts for the Peirce blocks \(E_i^*UE_j^*\), where \(U=T_{0,\omega}\), together with Krein-parameter arguments. In primitive cases, one obtains \(\dim T_{0,\omega}\in\{12,14,15\}\), while in imprimitive cases \(\dim T_{0,\omega}\in\{11,12\}\). The Krein-parameter test sharply restricts which primitive rank-\(3\) families can be triply-regular, and the subsequent comparison with \(\widetilde T_\omega\) 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-\(3\) permutation structure, and the incidence geometry of polar spaces, an alignment made explicit by the equality \(T_{0,\omega}=T_\omega=\widetilde T_\omega\) [2510.23441].

Source: https://www.emergentmind.com/topics/triply-transitive-strongly-regular-graphs