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Identifier

St001286: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 2

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

([],3) => 3

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

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

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

([],4) => 4

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

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

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

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

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

([(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) => 2

([],5) => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 3

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

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

([(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) => 2

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

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

([],6) => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The annihilation number of a graph.
For a graph on $m$ edges with degree sequence $d_1\leq\dots\leq d_n$, this is the largest number $k\leq n$ such that $\sum_{i=1}^k d_i \leq m$.

References

[1] Pepper, R. D. Binding independence MathSciNet:2706793

Code

def statistic(G):
    d = sorted(G.degree_iterator())
    m = G.num_edges()
    n = G.num_verts()
    a = 0
    for k in range(n):
        if a > m:
            return k-1
        a += d[k]
    if a > m:
        return n-1
    return n

Created

Nov 13, 2018 at 07:21 by Martin Rubey

Updated

Dec 29, 2018 at 22:57 by Martin Rubey