The source reports four affine systems passing every single-neighborhood marginal test constructively. Use exact multi-core constraints or a certificate-producing finite proof, and separately analyze the lower-core regime.
Route status · Narrowed routeAlgebraic graph theory · strongly regular graphs · finite configurations
The Missing Moore Graph
Collaboration betaA Moore graph of degree 57 and diameter 2 would have 3,250 vertices and extremal local structure: adjacent vertices share no common neighbor, while each nonadjacent pair shares exactly one. This last missing parameter case remains open.
Known results and sources
Research problem
Exact mathematical statement
Construct a Moore graph of degree and diameter , or prove that none exists. Equivalently, decide whether there exists a strongly regular graph with parameters
The retained source explicitly reports neither a construction nor a nonexistence proof.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on The Missing Moore Graph stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
Conditional on minimum degree three, the source census leaves a coordinate cross and the source then gives hand arguments for reversal and two chiral completions.
Evidence posture · Source-reported route statement · dependencies incompleteWork mapped so far
The Missing Moore Graph in numbers
4.2kretained lines of mathematical investigation4,194 in the current working snapshot
- Argument development
- 3,595 · 86%
- Explored or eliminated routes
- 72 · 2%
- Computational analysis
- 209 · 5%
- Open obligations
- 97 · 2%
- Definitions and setup
- 221 · 5%
6selected mapped statements1routes investigated6open questions6contribution-ready tasks
How this is measured
This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.
Recommended next task
Certify the minimum-degree-three step independently.
Suggested move: Obtain and verify a checkable pseudo-Boolean proof log, or replace the solver dependence with a hand elimination of the remaining degree-sequence cases.
What would count as progress
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
Argument map and routes
How the current approaches connect
Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.
Visible working map
Research route map
Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.
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Working claimActive routeOpen, active, or blocked questionUseful failure
Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.
Explored alternatives
Other routes
The source reports four affine systems passing every single-neighborhood marginal test constructively. Use exact multi-core constraints or a certificate-producing finite proof, and separately analyze the lower-core regime.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Certify the minimum-degree-three step independently.Suggested move: Obtain and verify a checkable pseudo-Boolean proof log, or replace the solver dependence with a hand elimination of the remaining degree-sequence cases.
Ready to work on
Resolve the surviving equality templates.Suggested move: Apply exact external-parent, intersection, saturation, and holonomy constraints across one or several coupled defect-zero cores.
Ready to work on
Exact construction-or-nonexistence target
The target is construction of a degree-57 diameter-2 Moore graph, equivalently srg(3250,57,0,1), or a nonexistence proof.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Minimum-degree certification gap
The exact OPB/model and infeasibility transcript support the no-degree-at-most-two assertion, but the source says a checkable certificate is missing.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Lower-core regime remains open
The κ≤33 regime is independently open and cannot be dropped even if the equality branch is eliminated.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Classify the lower-core regime.Suggested move: Analyze only κ≤33 signatures that survive the parameterized degree-product bounds and tuple caps, without assuming equality-branch behavior.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen problem
The degree-57 diameter-two Moore graph remains the last open case in the Hoffman–Singleton classification. A 2026 preprint proves that any such graph has no involutory automorphism, but supplies neither a construction nor a nonexistence proof.
[3]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintIshida proved that a hypothetical missing Moore graph has no involutions, hence has automorphism group of odd order; existence remains open.[3] Peer reviewedDalfó surveyed Moore graphs and retained the degree-57 graph of order 3250 as the missing case.[2] Peer reviewedHoffman and Singleton established the diameter-two Moore-graph classification framework that leaves the degree-57 parameter case.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusThe Missing Moore Graph
A degree-57 diameter-two Moore graph has the equivalent strongly regular parameters (3250,57,0,1).
[2][3]The Hoffman–Singleton graph is the existing degree-7 member of the diameter-two Moore-graph classification, not a solution of the degree-57 case.
[1]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetA formal statement of the exact degree-57 diameter-two Moore-graph existence problem and its strongly regular equivalence.
- Formalization targetFormal spectral and parameter feasibility theory for strongly regular graphs.
- Formalization targetCheckable encodings of any finite census or nonexistence certificate used in a future proof.
Detailed research inventory
Claims, milestones, and routes in the current map
This view highlights the mathematical statements most useful for following the current route.
4 standing statements2 proposed statements6 open questions1 narrowed routes
- theorem candidate
1 of 6 1 - reduction
2 of 6 2 - lemma
2 of 6 2 - computational claim
1 of 6 1
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.19 displayed rows · 1 route included
- retained route statementDoes an srg(3250,57,0,1) exist?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementDefect-zero cores have at most 34 edgesintermediate
- retained route statementRestricted census has exact recorded totalsintermediate
- retained route statementEquality branch reduces to a coordinate cross conditionallyintermediate
- Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
- DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
- Useful failureCurrent relaxationsreported failure
- Research targetCertify the minimum-degree-three step independently.open
- Research targetResolve the surviving equality templates.open
- Research targetClassify the lower-core regime.open
- Research targetExact construction-or-nonexistence targetopen
- Research targetMinimum-degree certification gapopen
- Research targetLower-core regime remains openopen
- Narrowed routeCurrent relaxationsThe source reports four affine systems passing every single-neighborhood marginal test constructively. Use exact multi-core constraints or a certificate-producing finite proof, and separately analyze the lower-core regime.
How to interpret these counts
A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.
Research outlook
Conditions that would advance the current route
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.
- Supply a complete argument with every imported premise identified.
- Survive an independent attempt to falsify the proposed step.
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Prepared starting pointCertify the minimum-degree-three step independently.
The Missing Moore Graph · ready to start
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Research contextPrepared context for any AI agent
A Moore graph of degree 57 and diameter 2 would have 3,250 vertices and extremal local structure: adjacent vertices share no common neighbor, while each nonadjacent pair shares exactly one. This last missing parameter case remains open.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
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Sources and references3 cited works · next context review by Nov 14, 2026
The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.
- 1On Moore graphs with diameters 2 and 3original source · Alan J. Hoffman, Robert R. Singleton · IBM Journal of Research and Development · 1960 · DOI 10.1147/rd.45.0497 · accessed Aug 14, 2026
- 2A survey on Moore graphssurvey or monograph · Cristina Dalfó · Linear Algebra and its Applications · 2019 · DOI 10.1016/j.laa.2018.12.035 · accessed Aug 14, 2026
- 3No involutions in the missing Moore graphpreprint · Yawara Ishida · arXiv · 2026-06-28 · ARXIV 2606.29183 · accessed Aug 14, 2026
Important qualifications
- The collection is a focused status and identity check, not an exhaustive survey of spectral, automorphism, or computational restrictions.
- The 2026 automorphism result is a preprint and does not construct the graph or prove nonexistence.
- No external finite computation or formalization was independently reproduced.
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