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Identifier

St000350: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

([(0,3),(1,2),(1,3),(2,3)],4) => 8

([(0,2),(0,3),(1,2),(1,3)],4) => 8

([(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) => 12

([],5) => 0

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

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

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

([(0,4),(1,4),(2,4),(3,4)],5) => 8

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

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

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

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

([(0,4),(1,4),(2,3),(3,4)],5) => 8

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

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

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

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

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

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

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

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

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

([(0,4),(1,3),(2,3),(2,4)],5) => 8

([(0,1),(2,3),(2,4),(3,4)],5) => 8

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

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

([(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) => 12

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

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

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

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

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 16

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

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 16

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

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

([],6) => 0

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

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

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

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

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 10

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

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

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

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

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

([(0,1),(2,5),(3,5),(4,5)],6) => 8

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

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 10

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 10

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

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

([(0,5),(1,5),(2,4),(3,4)],6) => 8

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 10

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 10

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 10

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 10

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 10

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

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

([(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) => 14

([(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) => 12

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

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

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 16

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 16

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

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

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

([(0,1),(2,5),(3,4),(4,5)],6) => 8

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

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 10

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 10

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 10

([(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) => 12

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

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 10

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

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

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

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Description

The sum of the vertex degrees of a graph.
This is clearly equal to twice the number of edges, and, incidentally, also equal to the trace of the Laplacian matrix of a graph. From this it follows that it is also the sum of the squares of the eigenvalues of the adjacency matrix of the graph.
The Laplacian matrix is defined as $D-A$ where $D$ is the degree matrix (the diagonal matrix with the vertex degrees on the diagonal) and where $A$ is the adjacency matrix. See [1] for detailed definitions.

References

[1] wikipedia:Laplacian_matrix

Code

def statistic(g):
    return 2*len(g.edges())

def statistic_alternative(g):
    return g.laplacian_matrix().trace()

Created

Dec 26, 2015 at 17:54 by Christian Stump

Updated

Feb 26, 2016 at 09:32 by Martin Rubey