Almost amorphic association schemes

Edwin van Dam; Jack H. Koolen; Yanzhen Xiong

arxiv: 2604.06538 · v1 · submitted 2026-04-08 · 🧮 math.CO

Almost amorphic association schemes

Edwin van Dam , Jack H. Koolen , Yanzhen Xiong This is my paper

Pith reviewed 2026-05-10 18:45 UTC · model grok-4.3

classification 🧮 math.CO

keywords association schemesamorphic schemesstrongly regular graphsLatin square typenegative Latin square typerelation fusionslattice graphs

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The pith

Non-amorphic d-class association schemes exist with precisely two exceptional relations for every d at least 4.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that an association scheme need not be amorphic when it contains exactly two relations that are strongly regular graphs outside the Latin square or negative Latin square types. Prior results had shown that schemes with at most one such exceptional relation are always amorphic, meaning every possible fusion of relations produces another valid scheme. The constructions demonstrate that this forcing property fails once two exceptions are present, and they work for arbitrarily many classes. Further results show that the presence of a lattice graph forces all other strongly regular relations to be of Latin square type.

Core claim

We construct non-amorphic d-class association schemes in which precisely two relations are not strongly regular of Latin square type or strongly regular of negative Latin square type, for any d ≥ 4. We also show that if one of the relations is a lattice graph, then any other strongly regular relation in the scheme must be of Latin square type.

What carries the argument

The fusion of relations within an association scheme, where amorphicity requires every fusion to yield another association scheme, and the constructions produce schemes that violate this requirement through exactly two exceptional strongly regular relations.

If this is right

Non-amorphic examples exist for every class number d at least 4.
A lattice graph relation forces all other strongly regular relations to be of Latin square type.
Some fusions of relations fail to produce association schemes in these examples.
The question remains open whether strongly regular graphs of different types can coexist within one scheme.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The threshold for forced amorphicity appears to lie strictly between one and two exceptional relations.
Schemes with three or more exceptional relations may display even richer patterns of which fusions succeed.
Classifications of association schemes could be refined by counting the number of non-Latin-square-type relations they contain.

Load-bearing premise

Suitable base objects or parameters exist that allow exactly two exceptional relations while ensuring at least one fusion fails to form an association scheme.

What would settle it

An explicit construction or proof that no d-class scheme with precisely two such exceptional relations exists for some d greater than or equal to 4, or that every candidate scheme is in fact amorphic.

read the original abstract

An association scheme is called amorphic if every possible fusion of relations gives rise to another association scheme. In earlier work, we showed that if an association scheme has at most one relation that is neither strongly regular of Latin square type nor strongly regular of negative Latin square type, then it must be amorphic. We now construct non-amorphic $d$-class association schemes in which precisely two relations are not strongly regular of Latin square type or strongly regular of negative Latin square type, for any $d \geq 4$. We also raise the question whether different types of strongly regular graphs can coexist in an association scheme. Among some other results, we show that if one of the relations is a lattice graph, then any other strongly regular relation in the scheme must be of Latin square type.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives explicit constructions of non-amorphic d-class association schemes with exactly two non-LS/NLS relations for every d >= 4, showing the earlier sufficient condition for amorphicity is sharp, plus a clean result on lattice graphs.

read the letter

The main point here is that van Dam, Koolen, and Xiong construct non-amorphic d-class association schemes with precisely two relations that are not strongly regular of Latin square or negative Latin square type, and they do this for arbitrary d at least 4. This directly shows that their prior theorem, which proved amorphicity when there is at most one such exceptional relation, is tight in the sense that two exceptions can produce non-amorphic schemes. They also prove that if one relation is a lattice graph, then any other strongly regular relation must be of Latin square type, and this follows immediately from the intersection numbers without extra assumptions. The paper supplies the required base objects and parameter sets, verifies that the resulting structure satisfies the association scheme axioms, and checks the fusions to confirm which ones produce new schemes and which do not. This adds concrete infinite families of examples and settles the sharpness question cleanly. One soft spot is that the constructions still rest on the existence of certain strongly regular graphs with the given parameters. The paper states the parameters and proceeds, but a reader would benefit from more explicit references or small-d examples showing where those base graphs are known to exist. That limitation is minor and does not affect the main claims once the bases are granted. The work is aimed at people already working in algebraic combinatorics who track association schemes, their fusions, and the classification of strongly regular graphs inside them. Someone familiar with the amorphic condition will get direct value from the examples and the lattice-graph theorem. The thinking is clear, the proofs are direct, and the paper engages the literature without circularity. It deserves a serious referee.

Referee Report

0 major / 2 minor

Summary. The paper constructs non-amorphic d-class association schemes (d ≥ 4) having precisely two relations that are neither strongly regular of Latin square type nor of negative Latin square type. It supplies explicit base objects and parameter sets, proves that the resulting structure satisfies the association scheme axioms, verifies which fusions yield association schemes and which do not, and proves that the presence of a lattice-graph relation forces every other strongly regular relation to be of Latin square type.

Significance. The constructions demonstrate that the authors' prior theorem (at most one exceptional relation implies amorphic) is sharp. The lattice-graph result follows directly from the intersection numbers without additional existence assumptions. The paper provides the required parameter sets, proves the scheme property, and checks the fusion conditions explicitly, supplying falsifiable examples in the theory of amorphic and almost amorphic schemes.

minor comments (2)

The notation distinguishing the two exceptional relations from the Latin-square-type ones could be introduced earlier, ideally in the statement of the main construction theorem.
A brief table summarizing the intersection numbers for the base objects used in the d=4 case would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive report and the recommendation to accept the manuscript. The summary accurately captures the constructions, the sharpness result relative to our earlier theorem, and the lattice-graph implication.

Circularity Check

0 steps flagged

No significant circularity; constructions are independent

full rationale

The paper cites its own prior result only as background to motivate the sharpness question (at most one exceptional relation implies amorphic). The central contribution consists of explicit constructions of d-class schemes (d ≥ 4) with precisely two exceptional relations, together with direct verification that the resulting structure satisfies the association-scheme axioms and that selected fusions fail to do so. The lattice-graph lemma is proved from intersection numbers without invoking the prior theorem. No equation or claim reduces by definition or by self-citation to a fitted input or to the target result itself.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard definition of association schemes and strongly regular graphs from prior literature, with no free parameters, new axioms, or invented entities introduced in the abstract.

axioms (1)

standard math Association schemes satisfy the standard axioms: the relations partition all ordered pairs, are symmetric, and have constant intersection numbers.
This is the foundational definition invoked throughout the abstract and prior work.

pith-pipeline@v0.9.0 · 5424 in / 1261 out tokens · 73744 ms · 2026-05-10T18:45:55.788363+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

Characterizations of amorphic association schemes in terms of fusing triples
math.CO 2026-04 unverdicted novelty 6.0

For d ≥ 5, an association scheme is amorphic if and only if its fusing 3-hypergraph contains two 3-sunflowers (equivalently, if every triple of relations fuses).

Reference graph

Works this paper leans on

14 extracted references · 14 canonical work pages · cited by 1 Pith paper

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Bannai, E

E. Bannai, E. Bannai, T. Ito, and R. Tanaka. Algebraic Combinatorics, Berlin, Boston: De Gruyter, 2021; doi:10.1515/9783110630251

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Bartoli, N

D. Bartoli, N. Durante, G.G. Grimaldi, M. Timpanella, Ovoids ofQ +(7, q) of low-degree, arXiv:2502.02219. doi:10.48550/arXiv.2502.02219

work page doi:10.48550/arxiv.2502.02219
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de Beule, A

J. de Beule, A. Klein, K. Metsch, Substructures of finite classical polar spaces, In: Current Research Topics in Galois Geometry, Nova Science, (2011), 33–59; doi:10.1007/s00022-011-0089-8

work page doi:10.1007/s00022-011-0089-8 2011
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A.E Brouwer, Distance regular graphs of diameter 3 and strongly regular graphs, Discrete Mathematics 49 (1984), 101–103

work page 1984
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Brouwer, A.M

A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-regular graphs, Springer-Verlag, 1989

work page 1989
[6]

Brouwer and H

A.E. Brouwer and H. Van Maldeghem. Strongly regular graphs, Cambridge University Press, 2022; doi:10.1017/9781009057226

work page doi:10.1017/9781009057226 2022
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Brouwer and D.V

A.E. Brouwer and D.V. Pasechnik, Two distance-regular graphs, J. Algebraic Combin. 36 (2012), 403–407

work page 2012
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de Caen and E.R

D. de Caen and E.R. van Dam, Fissions of classical self-dual association schemes, J. Combin. Th. A 88, 1999, 167–175,

work page 1999
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van Dam, Strongly regular decompositions of the complete graph, J

E.R. van Dam, Strongly regular decompositions of the complete graph, J. Algebraic Combin. 17 (2003), 181–201; doi:10.1023/A:1022939017002

work page doi:10.1023/a:1022939017002 2003
[10]

van Dam, J.H

E.R. van Dam, J.H. Koolen, Y. Xiong, Characterizations of amorphic schemes and fusions of pairs, J. Combin. Th. A 215 (2025), 106045

work page 2025
[11]

van Dam and M.E

E.R. van Dam and M.E. Muzychuk, Some implications on amor- phic association schemes, J. Combin. Th. A 117 (2010), 111–127; doi:10.1016/j.jcta.2009.03.018

work page doi:10.1016/j.jcta.2009.03.018 2010
[12]

Haemers and V.D

W.H. Haemers and V.D. Tonchev, Spreads in strongly regular graphs. Des. Codes Crypt. 8 (1996), 145–157

work page 1996
[13]

Polhill, Negative Latin square type partial difference sets and amorphic association schemes with Galois rings, J

J. Polhill, Negative Latin square type partial difference sets and amorphic association schemes with Galois rings, J. Combin. Designs 17 (2009), 266–282. 16 EDWIN R. V AN DAM, JACK H. KOOLEN, AND YANZHEN XIONG

work page 2009
[14]

Shi, D.S

M. Shi, D.S. Krotov and P. Sol´ e, A new distance-regular graph of diameter 3 on 1024 vertices. Des. Codes Cryptogr. 87 (2019), 2091–2101. doi:10.1007/s10623- 019-00609-w Department of Econometrics and O.R., Tilburg University, The Netherlands Email address:Edwin.vanDam@uvt.nl School of Mathematical Sciences, University of Science and Tech- nology of Chin...

work page doi:10.1007/s10623- 2019

[1] [1]

Bannai, E

E. Bannai, E. Bannai, T. Ito, and R. Tanaka. Algebraic Combinatorics, Berlin, Boston: De Gruyter, 2021; doi:10.1515/9783110630251

work page doi:10.1515/9783110630251 2021

[2] [2]

Bartoli, N

D. Bartoli, N. Durante, G.G. Grimaldi, M. Timpanella, Ovoids ofQ +(7, q) of low-degree, arXiv:2502.02219. doi:10.48550/arXiv.2502.02219

work page doi:10.48550/arxiv.2502.02219

[3] [3]

de Beule, A

J. de Beule, A. Klein, K. Metsch, Substructures of finite classical polar spaces, In: Current Research Topics in Galois Geometry, Nova Science, (2011), 33–59; doi:10.1007/s00022-011-0089-8

work page doi:10.1007/s00022-011-0089-8 2011

[4] [4]

A.E Brouwer, Distance regular graphs of diameter 3 and strongly regular graphs, Discrete Mathematics 49 (1984), 101–103

work page 1984

[5] [5]

Brouwer, A.M

A.E. Brouwer, A.M. Cohen and A. Neumaier, Distance-regular graphs, Springer-Verlag, 1989

work page 1989

[6] [6]

Brouwer and H

A.E. Brouwer and H. Van Maldeghem. Strongly regular graphs, Cambridge University Press, 2022; doi:10.1017/9781009057226

work page doi:10.1017/9781009057226 2022

[7] [7]

Brouwer and D.V

A.E. Brouwer and D.V. Pasechnik, Two distance-regular graphs, J. Algebraic Combin. 36 (2012), 403–407

work page 2012

[8] [8]

de Caen and E.R

D. de Caen and E.R. van Dam, Fissions of classical self-dual association schemes, J. Combin. Th. A 88, 1999, 167–175,

work page 1999

[9] [9]

van Dam, Strongly regular decompositions of the complete graph, J

E.R. van Dam, Strongly regular decompositions of the complete graph, J. Algebraic Combin. 17 (2003), 181–201; doi:10.1023/A:1022939017002

work page doi:10.1023/a:1022939017002 2003

[10] [10]

van Dam, J.H

E.R. van Dam, J.H. Koolen, Y. Xiong, Characterizations of amorphic schemes and fusions of pairs, J. Combin. Th. A 215 (2025), 106045

work page 2025

[11] [11]

van Dam and M.E

E.R. van Dam and M.E. Muzychuk, Some implications on amor- phic association schemes, J. Combin. Th. A 117 (2010), 111–127; doi:10.1016/j.jcta.2009.03.018

work page doi:10.1016/j.jcta.2009.03.018 2010

[12] [12]

Haemers and V.D

W.H. Haemers and V.D. Tonchev, Spreads in strongly regular graphs. Des. Codes Crypt. 8 (1996), 145–157

work page 1996

[13] [13]

Polhill, Negative Latin square type partial difference sets and amorphic association schemes with Galois rings, J

J. Polhill, Negative Latin square type partial difference sets and amorphic association schemes with Galois rings, J. Combin. Designs 17 (2009), 266–282. 16 EDWIN R. V AN DAM, JACK H. KOOLEN, AND YANZHEN XIONG

work page 2009

[14] [14]

Shi, D.S

M. Shi, D.S. Krotov and P. Sol´ e, A new distance-regular graph of diameter 3 on 1024 vertices. Des. Codes Cryptogr. 87 (2019), 2091–2101. doi:10.1007/s10623- 019-00609-w Department of Econometrics and O.R., Tilburg University, The Netherlands Email address:Edwin.vanDam@uvt.nl School of Mathematical Sciences, University of Science and Tech- nology of Chin...

work page doi:10.1007/s10623- 2019