- FindStat

Identifier

Mp00247: Graphs —de-duplicate⟶ Graphs
Mp00250: Graphs —clique graph⟶ Graphs
St001798: Graphs ⟶ ℤ

Values

([],1) => ([],1) => ([],1) => 0

([],2) => ([],1) => ([],1) => 0

([(0,1)],2) => ([(0,1)],2) => ([],1) => 0

([],3) => ([],1) => ([],1) => 0

([(1,2)],3) => ([(1,2)],3) => ([],2) => 0

([(0,2),(1,2)],3) => ([(0,1)],2) => ([],1) => 0

([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([],1) => 0

([],4) => ([],1) => ([],1) => 0

([(2,3)],4) => ([(1,2)],3) => ([],2) => 0

([(1,3),(2,3)],4) => ([(1,2)],3) => ([],2) => 0

([(0,3),(1,3),(2,3)],4) => ([(0,1)],2) => ([],1) => 0

([(0,3),(1,2)],4) => ([(0,3),(1,2)],4) => ([],2) => 0

([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => ([],2) => 0

([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,1)],2) => ([],1) => 0

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],1) => 0

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],1) => 0

([],5) => ([],1) => ([],1) => 0

([(3,4)],5) => ([(1,2)],3) => ([],2) => 0

([(2,4),(3,4)],5) => ([(1,2)],3) => ([],2) => 0

([(1,4),(2,4),(3,4)],5) => ([(1,2)],3) => ([],2) => 0

([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => ([],1) => 0

([(1,4),(2,3)],5) => ([(1,4),(2,3)],5) => ([],3) => 0

([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => ([(1,3),(2,3)],4) => 1

([(0,1),(2,4),(3,4)],5) => ([(0,3),(1,2)],4) => ([],2) => 0

([(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([],2) => 0

([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2)],3) => 0

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(1,3),(1,4),(2,3),(2,4)],5) => ([(1,2)],3) => ([],2) => 0

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([],2) => 0

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 1

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1)],2) => ([],1) => 0

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],1) => 0

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => ([],2) => 0

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 1

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1)],2) => 0

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 1

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3)],4) => 2

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 0

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 1

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],2) => 0

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1)],2) => 0

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1)],2) => 0

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],1) => 0

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],1) => 0

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],1) => 0

([],6) => ([],1) => ([],1) => 0

([(4,5)],6) => ([(1,2)],3) => ([],2) => 0

([(3,5),(4,5)],6) => ([(1,2)],3) => ([],2) => 0

([(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => ([],2) => 0

([(1,5),(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => ([],2) => 0

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1)],2) => ([],1) => 0

([(2,5),(3,4)],6) => ([(1,4),(2,3)],5) => ([],3) => 0

([(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => ([(1,3),(2,3)],4) => 1

([(1,2),(3,5),(4,5)],6) => ([(1,4),(2,3)],5) => ([],3) => 0

([(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([],2) => 0

([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => ([(1,3),(2,3)],4) => 1

([(0,1),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,2)],4) => ([],2) => 0

([(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2)],3) => 0

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2)],3) => 0

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(2,4),(2,5),(3,4),(3,5)],6) => ([(1,2)],3) => ([],2) => 0

([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,3),(1,2)],4) => ([],2) => 0

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => ([(1,3),(2,3)],4) => 1

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([],2) => 0

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 1

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2)],3) => 0

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 1

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,2)],3) => ([],2) => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,3),(1,2),(2,3)],4) => 1

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => 1

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([],2) => 0

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,2),(1,2)],3) => 1

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1)],2) => 0

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1)],2) => ([],1) => 0

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],1) => 0

([(0,5),(1,4),(2,3)],6) => ([(0,5),(1,4),(2,3)],6) => ([],3) => 0

([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,4),(2,3),(3,4)],5) => 1

([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,3),(2,3)],4) => 1

([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,4),(3,5),(4,5)],6) => ([],3) => 0

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 3

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,3),(2,3)],4) => 1

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2)],3) => 0

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 2

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,2)],3) => 0

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => 0

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 2

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 5

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 4

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4$

$F_{4} = 10 + q$

$F_{5} = 25 + 8\ q + q^{2}$

$F_{6} = 77 + 45\ q + 21\ q^{2} + 7\ q^{3} + 2\ q^{4} + 4\ q^{5}$

Description

The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph.
Let

$n(G)$ be the number of vertices of a graph

$G$ ,

$m(G)$ be its number of edges and let

$\alpha(G)$ be its independence number, St000093The cardinality of a maximal independent set of vertices of a graph.. Turán's theorem is that

$m(G) \geq m(T^c(n(G), \alpha(G)))$ where

$T^c(n, r)$ is the complement of the Turán graph.
This statistic records the difference.

Map

de-duplicate

Description

The de-duplicate of a graph.
Let

$G = (V, E)$ be a graph. This map yields the graph whose vertex set is the set of (distinct) neighbourhoods

$\{N_v | v \in V\}$ of

$G$ , and has an edge

$(N_a, N_b)$ between two vertices if and only if

$(a, b)$ is an edge of

$G$ . This is well-defined, because if

$N_a = N_c$ and

$N_b = N_d$ , then

$(a, b)\in E$ if and only if

$(c, d)\in E$ .
The image of this map is the set of so-called 'mating graphs' or 'point-determining graphs'.
This map preserves the chromatic number.

Map

clique graph

Description

The clique graph of a graph.
The clique graph of a graph

$G$ has as vertex set the set of maximal cliques

$G$ and an edge between vertices corresponding to cliques that intersect.
In other words, it is the intersection graph of the maximal cliques of

$G$ .