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Johnson graph: Graph-theoretic properties, Special cases & Johnson scheme

In mathematics, Johnson graphs are a special class of undirected graphs defined from systems of sets. The vertices of the Johnson graph J ( n , k ) {\displaystyle J(n,k)} are the k {\displaystyle k} -element subsets of an n {\displaystyle n} -element set; two vertices are adjacent when the intersection of the two vertices (subsets) contains ( k − 1 )…

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Johnson graph topic overview

The analysis highlights Graph-theoretic properties, Special cases and Johnson scheme as prominent areas in the source structure around Johnson graph.

Related topics
39
Source areas
8
Connected nodes
47
Extracted relationships
20
Related term clusters
35
Bridge connections
47

What this topic covers Research coverage

Source areas are shown by the number of related topics found in each part of the analysis. Use smaller areas too: they can reveal specialized angles and content gaps.

Graph-theoretic properties · 15 topics
Overview · 6 topics
Special cases · 6 topics
Johnson scheme · 5 topics
Automorphism group · 3 topics
Intersection array · 2 topics
Eigenvalues and eigenvectors · 1 topics
Open problems · 1 topics

Smaller areas are not necessarily less important. They contain fewer connections in this analysis and can be useful for finding specialized angles or coverage gaps.

Key facts & relationships

High-confidence facts extracted from structured source data. Use them as anchors for further research.

Diameter
min ( k , n − k ) {\displaystyle \min(k,n-k)}
Edges
1 2 k ( n − k ) ( n k ) {\displaystyle {\frac {1}{2}}k(n-k){\binom {n}{k}}}
Named after
Selmer M. Johnson
Notation
J ( n , k ) {\displaystyle J(n,k)}
Properties
k ( n − k ) {\displaystyle k(n-k)} -regular Vertex-transitive Distance-transitive Hamilton-connected Polytopal
Vertices
( n k ) {\displaystyle {\binom {n}{k}}}

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Explore all related topics Closing gaps

Browse the complete topic structure, not only the most central items. Less prominent entities and concepts can reveal missing angles, specialized context and useful research gaps. Each item opens a new analysis centered on that subject.

Overview

Special cases

Graph-theoretic properties

Automorphism group

Intersection array

Eigenvalues and eigenvectors

Johnson scheme

Open problems

For the semantics nerds

You can skip this section if you’re here for content ideas and keyword inspiration.

Advanced semantic analysis

How Johnson graph connects Entity context

The extracted context around Johnson graph shows recurring relationship patterns in the source. For example, Johnson graph → Each Johnson, Hamilton-connected, Hamiltonian, Johnson, N-k Another extracted example is Johnson graph → Johnson, K5, Kn, Kneser, Petersen. Use these groups to spot repeated connection types before inspecting the individual relationships.

Johnson graph

Top relations

related to Graph-theoretic properties · 5
Johnson graph → Each Johnson, Hamilton-connected, Hamiltonian, Johnson, N-k
related to Special cases · 5
Johnson graph → Johnson, K5, Kn, Kneser, Petersen
related to Johnson scheme · 2
Johnson graph → Johnson, The Johnson
Diameter · 1
Johnson graph → min ( k , n − k ) {\displaystyle \min(k,n-k)}
Edges · 1
Johnson graph → 1 2 k ( n − k ) ( n k ) {\displaystyle {\frac {1}{2}}k(n-k){\binom {n}{k}}}
Named after · 1
Johnson graph → Selmer M. Johnson
Notation · 1
Johnson graph → J ( n , k ) {\displaystyle J(n,k)}
Properties · 1
Johnson graph → k ( n − k ) {\displaystyle k(n-k)} -regular Vertex-transitive Distance-transitive Hamilton-connected Polytopal
Vertices · 1
Johnson graph → ( n k ) {\displaystyle {\binom {n}{k}}}
is a · 1
Johnson graph → open problem

Important terminology

Use these terms to understand the vocabulary surrounding the topic, not as a checklist for keyword stuffing.

Important terminology

displaystyle johnson graph graphs vertices scheme sets distance-transitive n-k number selmer intersection also properties pair given -element set related special

Johnson graph relationships Subject–Predicate–Object triples

TTTA extracted 20 structured relationships around Johnson graph. Examples in this analysis include Johnson graph → Diameter → min ( k , n − k ) {\displaystyle \min(k,n-k)} and Johnson graph → Edges → 1 2 k ( n − k ) ( n k ) {\displaystyle {\frac {1}{2}}k(n-k){\binom {n}{k}}}. The table shows each extracted connection, where it came from and its confidence.

SubjectPredicateObjectConfidenceSrc
Johnson graphDiametermin ( k , n − k ) {\displaystyle \min(k,n-k)}1.00infobox
Johnson graphEdges1 2 k ( n − k ) ( n k ) {\displaystyle {\frac {1}{2}}k(n-k){\binom {n}{k}}}1.00infobox
Johnson graphNamed afterSelmer M. Johnson1.00infobox
Johnson graphNotationJ ( n , k ) {\displaystyle J(n,k)}1.00infobox
Johnson graphPropertiesk ( n − k ) {\displaystyle k(n-k)} -regular Vertex-transitive Distance-transitive Hamilton-connected Polytopal1.00infobox
Johnson graphVertices( n k ) {\displaystyle {\binom {n}{k}}}1.00infobox
Johnson graphis aopen problem0.90text
Johnson graphrelated to Graph-theoretic propertiesHamilton-connected0.60section
Johnson graphrelated to Graph-theoretic propertiesHamiltonian0.60section
Johnson graphrelated to Graph-theoretic propertiesJohnson0.60section
Johnson graphrelated to Graph-theoretic propertiesN-k0.60section
Johnson graphrelated to Graph-theoretic propertiesEach Johnson0.60section

Related concept clusters Related term clusters

The concept neighborhoods around Johnson graph bring nearby vocabulary together. In this analysis, examples include Graph, Johnson and Graphs. Use the clusters to find adjacent concepts and terminology that may deserve separate research.

  • Johnson graph
    • Graph
    • Johnson
    • Graphs
    • Displaystyle
    • Scheme
    • Vertices
    • N-k
    • Sets
    • Intersection
    • Properties
    • Related
    • Selmer
  • johnson graph
    • Graph
    • Johnson
    • Displaystyle
    • Graphs
    • Scheme
    • Pair
    • Vertices
    • N-k
    • Sets
    • Intersection
    • Number
    • Properties
  • undirected graphs
    • Johnson
    • Diameter
    • Edges
    • Vertices
    • Properties
    • Related
    • Distance-transitive
    • Given
    • Scheme
    • Sets
    • Graph
    • Mathematics
  • johnson scheme
    • Graph
    • Graphs
    • Selmer
    • Displaystyle
    • Also
    • Pair
    • Scheme
    • Sets
    • Vertices
    • N-k
    • Array
    • Edges
  • selmer m. johnson
    • Graph
    • Graphs
    • Displaystyle
    • Scheme
    • Special
    • Vertices
    • N-k
    • Sets
    • Intersection
    • Properties
    • Related
    • Selmer
  • complete graph
    • Johnson
    • Displaystyle
    • Pair
    • N-k
    • Vertices
    • Intersection
    • Number
    • Scheme
    • Graphs
    • Array
    • Complement
    • Diameter
  • octahedral graph
    • Johnson
    • Displaystyle
    • Pair
    • N-k
    • Vertices
    • Intersection
    • Number
    • Scheme
    • Graphs
    • Array
    • Complement
    • Diameter
  • petersen graph
    • Johnson
    • Displaystyle
    • Pair
    • N-k
    • Vertices
    • Intersection
    • Number
    • Scheme
    • Graphs
    • Array
    • Complement
    • Diameter

Connections between topic areas Semantic bridges

For Johnson graph, one of the stronger structural bridges in this analysis connects Johnson graph with Graph-theoretic properties. Bridges highlight paths between different parts of the map and can reveal research angles that are easy to miss in a flat list.

Min side: 3
Johnson graph — Graph-theoretic properties · splits 32 ⟂ 16
Johnson graph — Overview · splits 41 ⟂ 7
Johnson graph — Special cases · splits 41 ⟂ 7
Johnson graph — Johnson scheme · splits 42 ⟂ 6
Johnson graph — Automorphism group · splits 44 ⟂ 4
Johnson graph — Intersection array · splits 45 ⟂ 3

Map overview Semantic statistics

Johnson graph

Nodes48
Edges47
Triples20
Avg. degree1.96
Density0.041667
Components1

Source & methodology

TTTA analyzes the structure around Johnson graph to surface related topics, entities, relationships, concept neighborhoods and bridge connections. Use the map to explore areas such as Graph-theoretic properties, Special cases & Johnson scheme, including less central topics that may reveal useful research gaps. Automatically extracted connections are research leads rather than rewritten encyclopedia content.

Source: Wikipedia — Johnson graph · EN edition · Analysis: TopicsToTalkAbout

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