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).
Core Idea¶
Moore graph is treated here as the recurring mathematicslogicstatistics 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 .
Scope of Application¶
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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.
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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.
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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 .
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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 ).
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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.
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).
Manages Complexity¶
Moore graph compresses multiple mathematicslogicstatistics 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.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics 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 ). 5.
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.
Relationships to Other Abstractions¶
Current abstraction Moore graph Domain-specific
Parents (1) — more general patterns this builds on
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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 (52nd 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