wiki_research

commit d860d9f746d3fa40daf0c2a36158f51069b25b2d
parent 6008a7d4c176a318c4b81529e7f6c0619a644738
Author: Antoine Amarilli <a3nm@a3nm.net>
Date:   Sat,  4 Jan 2025 15:30:53 +0100

commit with codex

Diffstat:
average_degree
 | 
5
+++++
crossing_number
 | 
11
+++++++++++
crossing_number_inequality
 | 
7
+++++++
degree
 | 
1
+
eulers_formula
 | 
9
+++++++++
mathematics_fields
 | 
1
+
planar_graph
 | 
15
+++++++++++----
statistics
 | 
6
++++++
theorem
 | 
2
++

9 files changed, 53 insertions(+), 4 deletions(-)
diff --git a/average_degree b/average_degree
@@ -0,0 +1,5 @@
+# Average degree
+
+The [average] of the [degree] of [vertices] in a [graph]
+
+Up: [degree], [average]
diff --git a/crossing_number b/crossing_number
@@ -0,0 +1,11 @@
+# Crossing number
+
+https://en.wikipedia.org/wiki/Crossing_number_(graph_theory)
+
+Lowest number of [edge] crossings of a [plane_embedding] of an [undirected_graph].
+
+An [undirected_graph] is a [planar_graph] iff the crossing number is zero
+
+[crossing_number_inequality]
+
+Up: [planar_graph]
diff --git a/crossing_number_inequality b/crossing_number_inequality
@@ -0,0 +1,7 @@
+# Crossing number inequality
+
+https://en.wikipedia.org/wiki/Crossing_number_inequality
+
+lower bound on the [crossing_number] as a function of the number of [edges] and [vertices]
+
+Up: [crossing_number], [inequality]
diff --git a/degree b/degree
@@ -5,5 +5,6 @@ Number of [neighbors] of a [vertex]
 In [directed_graphs], we can distinguish the [indegree] and [outdegree], respectively the number of [vertices] having a edge to the [vertex], or from the [vertex]
 
 - [maximal_degree]
+- [average_degree]
 
 Up: [graph_basic_notions]
diff --git a/eulers_formula b/eulers_formula
@@ -0,0 +1,9 @@
+# Euler's formula
+
+https://en.wikipedia.org/wiki/Euler_characteristic#Plane_graphs
+
+https://en.wikipedia.org/wiki/Planar_graph#Euler's_formula
+
+Up: [planar_graph]
+
+See also: [crossing_number_inequality]
diff --git a/mathematics_fields b/mathematics_fields
@@ -16,5 +16,6 @@ classification [msc]
 - [randomness]
 - [fourier_analysis]
 - [discrete_mathematics]
+- [statistics]
 
 Up: [mathematics]
diff --git a/planar_graph b/planar_graph
@@ -1,6 +1,6 @@
 # Planar graph
 
-A [graph_undirected] that can be embedded on [plane_mathematics]
+A [graph_undirected] that has a [plane_embedding], i.e., can be embedded on [plane_mathematics]
 
 - Recognizing them: [planarity_testing]
   - [dynamic_planarity_testing]
@@ -9,18 +9,25 @@ A [graph_undirected] that can be embedded on [plane_mathematics]
   - open for [treewidth]
 - [planar_language]
 
-[Theorem]s:
+[Theorems]:
 
 - [Kuratowskis_theorem] in terms of [topological_minors]
 - [Wagners_theorem] in terms of [graph_minors]
 
+Necessary conditions:
+
+- [Euler's_formula]
+  - implies that [average_degree] must be at most 6
+
 Specific algorithms:
 
 - [fkt_algorithm]
 
-Generalizations to [hypergraph]s: [planar_hypergraph]
+Generalizations:
 
-Generalization: [directed_planar_graph]
+- to [hypergraphs]: [planar_hypergraph]
+- to [directed_graphs]: [directed_planar_graph]
+- to quantify the "degree of non-planarity": [crossing_number]
 
 See also: [graph_drawing]
 
diff --git a/statistics b/statistics
@@ -0,0 +1,6 @@
+# Statistics
+
+- [average]
+- [median]
+
+Up: [mathematics_fields]
diff --git a/theorem b/theorem
@@ -37,3 +37,5 @@ A [result] in [mathematics], recognized as challenging to prove
 Up: [mathematics]
 
 See also: [automated_theorem_proving], [theoremkb]
+
+Aliases: theorems