- FindStat

Identifier

Mp00247: Graphs —de-duplicate⟶ Graphs
St000571: Graphs ⟶ ℤ

Values

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

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

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

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

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

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

([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => 24

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

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

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

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

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

([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 18

([(1,2),(1,3),(2,3)],4) => ([(1,2),(1,3),(2,3)],4) => 24

([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 44

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

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => 24

([(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) => 108

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

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

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

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

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

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

([(1,4),(2,3),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 18

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

([(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => 24

([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 18

([(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(2,4),(3,4)],5) => 44

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => 44

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

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(2,3)],4) => 18

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => 24

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 64

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => 44

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 24

([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 26

([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(2,3),(2,4),(3,4)],5) => 26

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 52

([(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) => 96

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 40

([(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) => 78

([(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) => 134

([(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) => 90

([(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) => 108

([(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) => 146

([(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) => 190

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => 44

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => 24

([(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) => 108

([(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) => 320

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

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

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

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

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

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

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

([(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => 18

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

([(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 24

([(1,5),(2,5),(3,4),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => 18

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

([(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 44

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 18

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 44

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 44

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

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

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(3,4)],5) => 18

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 26

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 24

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 64

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 18

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(2,3)],4) => 18

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,4),(3,4)],5) => 44

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 64

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => 44

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

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => 26

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => 18

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => 24

([(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) => 64

([(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) => 44

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

([(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) => 24

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

([(1,5),(2,4),(3,4),(3,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 26

([(0,1),(2,5),(3,4),(4,5)],6) => ([(0,1),(2,5),(3,4),(4,5)],6) => 20

([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(3,4),(3,5),(4,5)],6) => 26

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 46

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 52

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 46

([(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) => 90

([(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) => 96

([(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) => 158

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 40

([(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) => 72

([(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) => 78

([(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) => 60

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Description

The F-index (or forgotten topological index) of a graph.
This is
$$\sum_{v \in V(G)} d^3(v) = \sum_{\{u,v\}\in E(G)} \big(d^2(u)+d^2(v)\big)$$
where $d(u)$ is the degree of the vertex $u$.

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.