No involutions in the missing Moore graph

Yawara Ishida

arxiv: 2606.29183 · v1 · pith:EU36VQEZnew · submitted 2026-06-28 · 🧮 math.CO

No involutions in the missing Moore graph

Yawara Ishida This is my paper

Pith reviewed 2026-06-30 07:54 UTC · model grok-4.3

classification 🧮 math.CO

keywords Moore graphsdegree 57involutory automorphisms2-adic integersBrauer quotientspectral idempotentsautomorphism groupsHoffman-Singleton classification

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The pith

If a Moore graph of degree 57 exists, its automorphism group contains no involutions.

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

The paper shows that any Moore graph of degree 57 must lack automorphisms of order two. It reaches this by assuming an involution exists, using its known fixed-point counts, and producing a contradiction inside a module over the 2-adic integers. The contradiction arises when the ordinary trace of the involution on the summand cut out by the spectral idempotent for eigenvalue -8 fails to match the dimension of the corresponding Brauer quotient. This restriction tightens the possible symmetry groups for the last unresolved Moore graph of diameter two.

Core claim

A Moore graph of degree 57 has no involutory automorphisms. The proof proceeds by considering the vertex module over the ring of 2-adic integers, extracting the direct summand given by the image of the spectral idempotent for the eigenvalue -8, and comparing the ordinary trace of the involution on this summand with the dimension of its Brauer quotient; the mismatch, when combined with the known fixed-point structure, yields the desired contradiction.

What carries the argument

The 2-adic vertex module and its direct summand for the -8 eigenspace, whose trace versus Brauer-quotient dimension comparison supplies the obstruction.

If this is right

The automorphism group of any such graph has no elements of order 2.
All further known restrictions on the automorphism group must be compatible with an odd-order group.
Any construction or non-existence proof for the graph can restrict attention to odd-order symmetries.

Where Pith is reading between the lines

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

The same trace-Brauer comparison might rule out involutions on other strongly regular graphs whose eigenvalues allow a similar 2-adic splitting.
Existence questions for Moore graphs may ultimately reduce to checking only odd-order automorphism groups.
If the graph exists, its adjacency algebra over the 2-adics must admit no fixed summands under involutions of the predicted type.

Load-bearing premise

The fixed-point counts that any involution on a degree-57 Moore graph must satisfy are taken as given from earlier work.

What would settle it

An explicit involution on a hypothetical degree-57 Moore graph whose action on the 2-adic summand makes the ordinary trace equal the Brauer-quotient dimension would falsify the claim.

read the original abstract

The Moore graph of degree $57$, if one exists, is the remaining open case of the Hoffman-Singleton classification in diameter two. Although its existence remains open, substantial restrictions on the automorphism group of such a graph are known. In this paper we prove that a Moore graph of degree $57$ has no involutory automorphisms. The proof combines the known fixed-point structure of an involution with a module-theoretic obstruction. More precisely, we consider the vertex module over the ring of 2-adic integers and the direct summand given by the image of the spectral idempotent for the eigenvalue $-8$. Comparing the ordinary trace of the involution on this summand with the dimension of its Brauer quotient gives a contradiction.

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 shows that the degree-57 Moore graph has no involutory automorphisms by getting a contradiction between the trace on a 2-adic summand and the dimension of its Brauer quotient.

read the letter

The main point is that this paper rules out involutions as automorphisms of a hypothetical Moore graph of degree 57. It takes the known fixed-point counts for an involution and runs them through a 2-adic vertex module, isolating the summand coming from the spectral idempotent for eigenvalue -8, then compares the ordinary trace against the Brauer quotient dimension to reach a numerical contradiction.

The module calculation is the part that looks new. Prior work already limited the automorphism group, but this specific obstruction using the 2-adic structure and Brauer quotient on that eigenspace does not appear in the earlier references. The argument is direct and avoids any parameter fitting or self-referential definitions.

The reliance on the imported fixed-point structure is the clearest soft spot, but that is standard in this area and is flagged explicitly. If the trace-versus-dimension step checks out in the full write-up, the rest follows without circularity. No internal inconsistencies show up from the description.

People working on Moore graphs, strongly regular graphs, or algebraic constraints on automorphism groups will find this useful as one more concrete restriction. It does not settle existence but narrows the symmetry possibilities. The work shows clear engagement with the literature and the relevant module theory, so it deserves a serious referee to verify the calculation details.

Referee Report

0 major / 1 minor

Summary. The manuscript proves that a Moore graph of degree 57 (if it exists) admits no involutory automorphisms. The argument invokes the known fixed-point structure of an involution on such a graph, then obtains a contradiction by considering the vertex module over the 2-adic integers, the direct summand cut out by the spectral idempotent for eigenvalue −8, and comparing the ordinary trace of the involution on this summand with the dimension of its Brauer quotient.

Significance. The result supplies a further restriction on the automorphism group of the missing Moore graph, complementing earlier work on the Hoffman–Singleton classification. The combination of combinatorial fixed-point data with a 2-adic module calculation (ordinary trace versus Brauer quotient dimension) is a clean, parameter-free derivation that relies only on standard eigenvalue information and prior fixed-point theorems; this methodological approach may be reusable for other questions in algebraic graph theory.

minor comments (1)

[Abstract] The abstract states that the fixed-point structure is 'known' but does not cite the precise reference in the first sentence; adding the citation there would improve immediate readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, accurate summary of the proof, and recommendation to accept. No major comments require a point-by-point reply.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation proceeds from the externally cited fixed-point structure of an involution (explicitly flagged as prior literature) to a contradiction via the 2-adic vertex module, the spectral idempotent summand for eigenvalue -8, ordinary trace, and Brauer quotient dimension. No equation or step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the central numerical obstruction is independent of the target claim and uses standard module-theoretic comparisons. This is the most common honest finding for a self-contained proof against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof rests on the known fixed-point structure of involutions in Moore graphs and on standard facts about the spectrum and the 2-adic module; no free parameters or new entities are introduced.

axioms (2)

domain assumption Known fixed-point structure of an involution on a Moore graph of degree 57
Invoked explicitly in the abstract as the starting point of the argument.
domain assumption Existence and properties of the spectral idempotent for eigenvalue -8
Used to define the direct summand inside the 2-adic vertex module.

pith-pipeline@v0.9.1-grok · 5639 in / 1103 out tokens · 26494 ms · 2026-06-30T07:54:20.462855+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

13 extracted references · 6 canonical work pages

[1]

Bannai and T

E. Bannai and T. Ito,On finite Moore graphs, J. Fac. Sci. Univ. Tokyo Sect. IA Math.20(1973), 191–208. https://repository.dl. itc.u-tokyo.ac.jp/record/39795/files/jfs200201.pdf

1973
[2]

Bouc and J

S. Bouc and J. Thévenaz,The primitive idempotents of thep-permutation ring, J. Algebra323(2010), no. 10, 2905–2915.https://doi.org/10. 1016/j.jalgebra.2009.11.036

2010
[3]

Broué,On Scott modules andp-permutation modules: an approach through the Brauer morphism, Proc

M. Broué,On Scott modules andp-permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc.93(1985), no. 3, 401–408.https://doi.org/10.1090/S0002-9939-1985-0773988-9

work page doi:10.1090/s0002-9939-1985-0773988-9 1985
[4]

A. E. Brouwer and W. H. Haemers,Spectra of Graphs, Uni- versitext, Springer, New York, 2012. https://doi.org/10.1007/ 978-1-4614-1939-6

2012
[5]

P. J. Cameron,Permutation Groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, Cambridge, 1999. https://doi.org/10.1017/CBO9780511623677

work page doi:10.1017/cbo9780511623677 1999
[6]

Dalfó,A survey on the missing Moore graph, Linear Algebra Appl

C. Dalfó,A survey on the missing Moore graph, Linear Algebra Appl. 569(2019), 1–14.https://doi.org/10.1016/j.laa.2018.12.035

work page doi:10.1016/j.laa.2018.12.035 2019
[7]

R. M. Damerell,On Moore graphs, Math. Proc. Cambridge Phi- los. Soc.74(1973), no. 2, 227–236. https://doi.org/10.1017/ S0305004100048015 9

1973
[8]

A. J. Hoffman and R. R. Singleton,On Moore graphs with diameters 2 and 3, IBM J. Res. Develop.4(1960), 497–504.https://doi.org/10. 1147/rd.45.0497

1960
[9]

Lassueur,A tour of p-permutation modules and related classes of modules, Jahresber

C. Lassueur,A tour of p-permutation modules and related classes of modules, Jahresber. Dtsch. Math.-Ver.125(2023), 137–189. https: //doi.org/10.1365/s13291-023-00266-y

work page doi:10.1365/s13291-023-00266-y 2023
[10]

Mačaj and J

M. Mačaj and J. Širáň,Search for properties of the missing Moore graph, Linear Algebra Appl.432(2010), no. 9, 2381–2398.https://doi.org/ 10.1016/j.laa.2009.07.018

work page doi:10.1016/j.laa.2009.07.018 2010
[11]

A. A. Makhnev and D. V. Paduchikh,Automorphisms of Aschbacher graphs, Algebra and Logic40(2001), no. 2, 69–74; translated from Algebra i Logika40(2001), no. 2, 125–134.https://doi.org/10.1023/ A:1010217919915

2001
[12]

A. A. Makhnev and D. V. Paduchikh,On the automorphism group of the Aschbacher graph, Proc. Steklov Inst. Math.267(2009), Suppl. 1, 149–163.https://doi.org/10.1134/S0081543809070141

work page doi:10.1134/s0081543809070141 2009
[13]

Miller and J

M. Miller and J. Širáň,Moore graphs and beyond: a survey of the degree/diameter problem, Electron. J. Combin.20(2013), no. 2, Dynamic Survey DS14v2. https://www.combinatorics.org/files/ Surveys/ds14/ds14v2-2013.pdf 10

2013

[1] [1]

Bannai and T

E. Bannai and T. Ito,On finite Moore graphs, J. Fac. Sci. Univ. Tokyo Sect. IA Math.20(1973), 191–208. https://repository.dl. itc.u-tokyo.ac.jp/record/39795/files/jfs200201.pdf

1973

[2] [2]

Bouc and J

S. Bouc and J. Thévenaz,The primitive idempotents of thep-permutation ring, J. Algebra323(2010), no. 10, 2905–2915.https://doi.org/10. 1016/j.jalgebra.2009.11.036

2010

[3] [3]

Broué,On Scott modules andp-permutation modules: an approach through the Brauer morphism, Proc

M. Broué,On Scott modules andp-permutation modules: an approach through the Brauer morphism, Proc. Amer. Math. Soc.93(1985), no. 3, 401–408.https://doi.org/10.1090/S0002-9939-1985-0773988-9

work page doi:10.1090/s0002-9939-1985-0773988-9 1985

[4] [4]

A. E. Brouwer and W. H. Haemers,Spectra of Graphs, Uni- versitext, Springer, New York, 2012. https://doi.org/10.1007/ 978-1-4614-1939-6

2012

[5] [5]

P. J. Cameron,Permutation Groups, London Mathematical Society Student Texts, vol. 45, Cambridge University Press, Cambridge, 1999. https://doi.org/10.1017/CBO9780511623677

work page doi:10.1017/cbo9780511623677 1999

[6] [6]

Dalfó,A survey on the missing Moore graph, Linear Algebra Appl

C. Dalfó,A survey on the missing Moore graph, Linear Algebra Appl. 569(2019), 1–14.https://doi.org/10.1016/j.laa.2018.12.035

work page doi:10.1016/j.laa.2018.12.035 2019

[7] [7]

R. M. Damerell,On Moore graphs, Math. Proc. Cambridge Phi- los. Soc.74(1973), no. 2, 227–236. https://doi.org/10.1017/ S0305004100048015 9

1973

[8] [8]

A. J. Hoffman and R. R. Singleton,On Moore graphs with diameters 2 and 3, IBM J. Res. Develop.4(1960), 497–504.https://doi.org/10. 1147/rd.45.0497

1960

[9] [9]

Lassueur,A tour of p-permutation modules and related classes of modules, Jahresber

C. Lassueur,A tour of p-permutation modules and related classes of modules, Jahresber. Dtsch. Math.-Ver.125(2023), 137–189. https: //doi.org/10.1365/s13291-023-00266-y

work page doi:10.1365/s13291-023-00266-y 2023

[10] [10]

Mačaj and J

M. Mačaj and J. Širáň,Search for properties of the missing Moore graph, Linear Algebra Appl.432(2010), no. 9, 2381–2398.https://doi.org/ 10.1016/j.laa.2009.07.018

work page doi:10.1016/j.laa.2009.07.018 2010

[11] [11]

A. A. Makhnev and D. V. Paduchikh,Automorphisms of Aschbacher graphs, Algebra and Logic40(2001), no. 2, 69–74; translated from Algebra i Logika40(2001), no. 2, 125–134.https://doi.org/10.1023/ A:1010217919915

2001

[12] [12]

A. A. Makhnev and D. V. Paduchikh,On the automorphism group of the Aschbacher graph, Proc. Steklov Inst. Math.267(2009), Suppl. 1, 149–163.https://doi.org/10.1134/S0081543809070141

work page doi:10.1134/s0081543809070141 2009

[13] [13]

Miller and J

M. Miller and J. Širáň,Moore graphs and beyond: a survey of the degree/diameter problem, Electron. J. Combin.20(2013), no. 2, Dynamic Survey DS14v2. https://www.combinatorics.org/files/ Surveys/ds14/ds14v2-2013.pdf 10

2013