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Identifier

St001827: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 1

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 10

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 15

([],5) => 0

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

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

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

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

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

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

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

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

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

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

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 9

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

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 10

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 8

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 9

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

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

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 32

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

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

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 9

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

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 10

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 19

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 32

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

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 16

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 31

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 51

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 33

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 77

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 110

([],6) => 0

([(4,5)],6) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 12

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

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

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

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

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

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

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

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 14

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 8

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 12

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

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 12

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 14

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 32

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

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 52

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

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 96

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

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

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

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

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

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

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

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 12

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 9

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 27

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

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 14

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 11

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 15

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Description

The number of two-component spanning forests of a graph.
A spanning subgraph is a subgraph which contains all vertices of the ambient graph. A forest is a graph which contains no cycles, and has any number of connected components. A two-component spanning forest is a spanning subgraph which contains no cycles and has two connected components.

References

[1] Kassel, A., Kenyon, R., Wu, W. Random two-component spanning forests MathSciNet:3414453 zbMATH:1334.82011 DOI:10.1214/14-AIHP625 arXiv:1203.4858
[2] Number of forests with two connected components in the complete graph K_n. OEIS:A083483
[3] Number a(n) of forests with two components in the complete bipartite graph K_n,n. OEIS:A100070

Code

def statistic(g):
    n = len(g.vertices())
    L = g.kirchhoff_matrix()
    # reduced Laplacian matrix
    Lred = L[:n-1,:n-1]
    a = 0
    for i in range(n-1):
        skip_i = [j for j in range(n-1) if j != i]
        # add number of forests rooted at i and n-1
        a += Lred[skip_i,skip_i].det()
    for (i,j,_) in g.edges():
        if i != n-1 and j != n-1 and i != j:
            skip_ij = [k for k in range(n-1) if k != i and k != j]
            # subtract number of forests rooted at e+, e-, and n-1
            a -= Lred[skip_ij, skip_ij].det()
    return a

Created

Jul 26, 2022 at 11:37 by Harry Richman

Updated

Jul 26, 2022 at 11:37 by Harry Richman