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Identifier

St001277: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

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

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

([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => 0

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

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

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

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

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

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

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

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

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

([(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) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 2

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

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

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

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

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

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

([(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) => 2

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + 2\ q + q^{2}$

$F_{4} = 1 + 5\ q + 4\ q^{2} + q^{3}$

$F_{5} = 1 + 9\ q + 18\ q^{2} + 5\ q^{3} + q^{4}$

$F_{6} = 1 + 19\ q + 85\ q^{2} + 43\ q^{3} + 7\ q^{4} + q^{5}$

Description

The degeneracy of a graph.
The degeneracy of a graph

$G$ is the maximum of the minimum degrees of the (vertex induced) subgraphs of

$G$ .

References

[1] wikipedia:Degeneracy_(graph_theory)

Code

def statistic(G):
    d = []
    H = G.copy(immutable=False)
    while H.num_verts():
        d_min = None
        for v in H.vertices():
            d_v = H.degree(v)
            if d_min is None:
                d_min = d_v
                w = v
            elif d_min > d_v:
                d_min = d_v
                w = v
        d.append(d_min)
        H.delete_vertex(w)
    return max(d)

def statistic(G):
    k = 0
    while G.cores(k)[1]:
        k += 1
    return k-1

Created

Oct 23, 2018 at 11:38 by Martin Rubey

Updated

Oct 23, 2018 at 11:38 by Martin Rubey