Moore graph¶

In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices).

Version: v1 · 2026-09-28 · History
Domain-specific #: 10809
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Graph Theory, Extremal Graph Theory → Mathematics

Core Idea¶

Moore graph is treated here as the recurring mathematics_logic_statistics identity summarized by this source-grounded definition: In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices).

In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). If the degree of such a graph is and its diameter is , its girth must equal . This is true, for a graph of degree and diameter , if and only if its number of vertices (its order) equals.

1 + d\sum_{i=0}^{k-1}(d-1)i,. an upper bound on the largest possible number of vertices in any graph with this degree and diameter. Therefore, these graphs solve the degree diameter problem for their parameters.

For Moore graph, the abstraction is narrower than the article's general subject matter: a positive case must preserve In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). Retaining only the name, a familiar example, or a downstream effect is insufficient. The specialist roles and tests remain anchored in mathematics_logic_statistics, which is why this identity is domain-specific rather than prime.

Structural Signature¶

Sig role-phrases:

Defining carrier — Let be any graph with maximum degree and diameter , and consider the tree formed by breadth-first search starting from any vertex .
Constitutive relation — If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons.
Operating condition — Moore graphs were named by after Edward F.
Recognition evidence — This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ).
Admissible variation — In the next level, there are at most vertices: each neighbor of uses one of its adjacencies to connect to and so can have at most neighbors at level 2.
Characteristic consequence — In general, a similar argument shows that at any level , there can be at most vertices.
Failure boundary — originally defined a Moore graph as a graph for which this bound on the number of vertices is met exactly.

What It Is Not¶

Not the whole field of mathematics_logic_statistics. The node requires the specific identity stated by In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices).
Not an over-broad reading. Each Moore graph has girth exactly : it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first levels of some breadth-first search tree.
Not an over-broad reading. Let be any graph with maximum degree and diameter , and consider the tree formed by breadth-first search starting from any vertex .
Not an over-broad reading. This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ).
Not automatically Degree diameter problem. Retrieval proximity does not establish equivalence; the two identities must be compared by carrier, operation, and failure boundary.

Scope of Application¶

Moore graph applies literally inside mathematics_logic_statistics wherever the source-defined carrier and relation can be established. Its documented habitats include:

Examples. If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons.
Moore graphs as cages. Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its girth.
Bounding vertices by degree and diameter. Let be any graph with maximum degree and diameter , and consider the tree formed by breadth-first search starting from any vertex .
Bounding vertices by degree and diameter. This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ).
Bounding vertices by degree and diameter. In the next level, there are at most vertices: each neighbor of uses one of its adjacencies to connect to and so can have at most neighbors at level 2.
Bounding vertices by degree and diameter. In general, a similar argument shows that at any level , there can be at most vertices.

Outside mathematics_logic_statistics, the name should be retained only when these same operational conditions survive; otherwise the comparison belongs to the broader parent Pattern or should be marked as analogy.

Clarity¶

A clear use of Moore graph names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). The strongest recognition evidence in the frozen account is: This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ). A report should distinguish that evidence from a proxy, consequence, or common implementation. It should also state the qualification Each Moore graph has girth exactly : it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first levels of some breadth-first search tree. so that a reader can reproduce the classification rather than infer it from topical resemblance.

Manages Complexity¶

Moore graph compresses multiple mathematics_logic_statistics details into a stable diagnostic relation. The source shows both the central mechanism—if the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons.—and the practical consequence—in general, a similar argument shows that at any level , there can be at most vertices. This compression makes cases comparable while leaving parameters, conventions, exceptions, and evidential quality explicit. It is lossy by design: local history and implementation details may be omitted only when they do not alter the defining relation.

Abstract Reasoning¶

Type the carrier. Identify the mathematics_logic_statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices).
Check operation and conditions. Moore graphs were named by after Edward F.
Demand recognition evidence. This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ).
Test variation. Change an implementation or setting while preserving in the next level, there are at most vertices: each neighbor of uses one of its adjacencies to connect to and so can have at most neighbors at level 2.
Run the collapse test. Remove the defining operation; if the label still seems equally apt, only a topic or correlate was retained.
Reduce cautiously. When the specialist conditions cannot be carried, route the residual comparison to Pattern.

Knowledge Transfer¶

Within the home domain. Knowledge about Moore graph transfers literally when a new case preserves the same carrier type, relation, and recognition test. If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons. Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its girth.

Beyond the home domain. No canonical parent is asserted for Moore graph. An outside case receives the specialist name only when the same typed roles and rejection conditions can be filled literally; otherwise the comparison remains an analogy pending later graph densification.

Examples¶

Canonical¶

Thus, the Moore graphs are sometimes defined as including the graphs that exactly meet this bound. This case is canonical because it supplies a concrete carrier and lets the defining relation be checked rather than merely named.

Mapped back: carrier → the entities in the documented case; operation → In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices); recognition evidence → This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of )

Applied / In Practice¶

The even girth case also follows from the Feit-Higman theorem about possible values of for a generalized -gon. The applied case shows how the identity is used under a second setting or qualification while keeping the same operative relation.

Mapped back: changed setting → Examples; invariant → In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices); boundary → the case exits the class when each Moore graph has girth exactly : it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first levels of some breadth-first search tree

Structural Tensions¶

T1 — Stable identity versus admissible variation. Each Moore graph has girth exactly : it does not have enough vertices to have higher girth, and a shorter cycle would cause there to be too few vertices in the first levels of some breadth-first search tree. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Which changes preserve the defining relation, and which replace it?

T2 — Recognition versus proxy. Let be any graph with maximum degree and diameter , and consider the tree formed by breadth-first search starting from any vertex . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the cited evidence establish the identity or only a correlated sign?

T3 — Definition versus implementation. This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ). The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Is the observed implementation constitutive, optional, or merely common?

T4 — Scope versus overextension. In the next level, there are at most vertices: each neighbor of uses one of its adjacencies to connect to and so can have at most neighbors at level 2. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Can every claimed application fill the same typed roles without metaphor?

T5 — Transfer versus domain accent. Let be any graph with maximum degree and diameter , and consider the tree formed by breadth-first search starting from any vertex . The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: Does the receiving case instantiate Moore graph literally, co-instantiate Pattern, or only resemble it?

T6 — Autonomy versus reduction. If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons. The tension matters because emphasizing only one side either dissolves the identity or overstates what the evidence and domain conventions warrant.

Diagnostic: What does Moore graph distinguish that the broader parent Pattern leaves together?

Structural–Framed Character¶

Moore graph is structural-leaning. Its structural side is the repeatable organization summarized by In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). Its framed side is the mathematics_logic_statistics vocabulary that fixes the carrier, evidence, exceptions, and admissible transformations.

Evaluative weight: the identity can be stated descriptively even when applications carry practical stakes. Human-practice dependence: the source-grounded carrier determines whether the relation exists independently or is constituted by a practice. Institutional origin: disciplinary conventions stabilize the name and test. Vocabulary portability: Moore graphs were named by after Edward F. Import versus recognition: literal transfer requires the same mechanism; shape alone is analogy.

Its portable skeleton is Pattern. Its character: a recurring specialist identity whose thin organization can be abstracted, while its operational meaning remains domain-bound.

Structural Core vs. Domain Accent¶

What is skeletal. In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). The stable skeleton is the typed relation expressed in that definition and the entry's recognition and collapse tests. The source identifies these operative conditions: Let be any graph with maximum degree and diameter , and consider the tree formed by breadth-first search starting from any vertex . If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons. It further constrains recognition and variation through: Moore graphs were named by after Edward F. This tree has 1 vertex at level 0 ( itself), and at most vertices at level 1 (the neighbors of ).

What is domain-bound. mathematics logic statistics supplies the operative entities, technical vocabulary, warrants, and exceptions that make Moore graph literal. Its documented scope includes the condition that If the generalized definition of Moore graphs that allows even girth graphs is used, the even girth Moore graphs correspond to incidence graphs of (possible degenerate) generalized polygons. Another bounded application condition is that Instead of upper bounding the number of vertices in a graph in terms of its maximum degree and its diameter, we can calculate via similar methods a lower bound on the number of vertices in terms of its minimum degree and its girth. These are not decorative examples; they determine which carrier and evidence can fill the abstraction's roles.

Why no parent is asserted. Removing those specialist details does not currently yield one live catalog node that is a necessary genus for every instance. The entry is therefore approved as unparented rather than attached by topical resemblance. Its collapse evidence remains specific—In the next level, there are at most vertices: each neighbor of uses one of its adjacencies to connect to and so can have at most neighbors at level 2.—and future graph densification may discover a defensible relation only if it preserves that boundary.

This entry is a kind of Network.

Approved unparented node. No current live node supplies a defensible necessary genus or structural prerequisite for Moore graph. The reviewed identity is: In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices). The accelerated suggestion was declined because topical or lexical similarity does not establish hierarchy; the node is admitted without a parent pending later graph densification.
Related reasoning operations. Evidence, representation, comparison, classification, transformation, or evaluation may participate in particular cases, but participation does not make any one of them a necessary parent of every instance.

Relationships to Other Abstractions¶

Current abstraction Moore graph Domain-specific

Parents (1) — more general patterns this builds on

Moore graph is a kind of Network Prime

A Moore graph is a graph/network attaining the degree-diameter Moore bound; Graph is a declared alias of the live Network Prime.

Hierarchy path (1) — routes to 1 parentless root

Moore graph → Network → Reservoir-Flux Network → Conservation Laws → Invariance

Neighborhood in Abstraction Space¶

Moore graph sits in a moderately populated region (52^nd percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Data Structures & Graph Variants (17 abstractions)

Nearest neighbors

Maximal independent set — 0.87
Block Graph — 0.86
Complement graph — 0.86
2–3 Heap — 0.85
Graph Toughness — 0.85

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With¶

Pattern. The parent omits the specialist differentia. Tell: Can the case establish In graph theory, a Moore graph is a regular graph whose girth (the shortest cycle length) is more than twice its diameter (the distance between the farthest two vertices)?
Degree diameter problem. Maximize the number of vertices in a finite graph subject to simultaneous maximum-degree and diameter bounds, comparing constructions with the breadth-first Moore upper bound. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Cage (Graph Theory). An r-regular graph of girth g having the minimum possible number of vertices among all graphs with that degree and girth. Tell: Which entry's carrier, operation, and failure condition are satisfied?
Girth (Graph Theory). Assign an undirected graph the length of its shortest cycle, using infinity for an acyclic graph, to quantify how far local neighborhoods remain tree-like. Tell: Which entry's carrier, operation, and failure condition are satisfied?
A measurement, proxy, or consequence. Those may provide evidence without being the identity. Tell: Would Moore graph remain present if the detector or downstream effect changed?
A metaphorical analogue. A similar shape outside mathematics_logic_statistics lacks the specialist mechanism. Tell: Do the native roles transfer literally, or only the parent Pattern?

References¶

Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/Moore_graph (revision 1369968557).
Preserved source candidate: http://repository.dl.itc.u-tokyo.ac.jp/dspace/handle/2261/6123
Preserved source candidate: https://web.archive.org/web/20120424231252/http://repository.dl.itc.u-tokyo.ac.jp/dspace/handle/2261/6123
Preserved source candidate: https://upcommons.upc.edu/bitstream/2117/127212/1/monster%2818oct2017-rev30dec2018%29.pdf
Preserved source candidate: http://www.math-inst.hu/~p_erdos/1966-06.pdf
Preserved source candidate: https://web.archive.org/web/20160309214909/http://www.math-inst.hu/~p_erdos/1966-06.pdf
Preserved source candidate: http://homepages.cwi.nl/~aeb/math/ipm/ipm.pdf

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.