Identifier
Identifier
Values
([],1) generating graphics... => 0
([],2) generating graphics... => 0
([(0,1)],2) generating graphics... => 1
([],3) generating graphics... => 0
([(1,2)],3) generating graphics... => 1
([(0,1),(0,2),(1,2)],3) generating graphics... => 2
([],4) generating graphics... => 0
([(2,3)],4) generating graphics... => 1
([(1,2),(1,3),(2,3)],4) generating graphics... => 2
([(0,3),(1,2)],4) generating graphics... => 1
([(0,2),(0,3),(1,2),(1,3)],4) generating graphics... => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) generating graphics... => 3
([],5) generating graphics... => 0
([(3,4)],5) generating graphics... => 1
([(2,3),(2,4),(3,4)],5) generating graphics... => 2
([(0,4),(1,4),(2,4),(3,4)],5) generating graphics... => 2
([(1,4),(2,3)],5) generating graphics... => 1
([(1,3),(1,4),(2,3),(2,4)],5) generating graphics... => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 3
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) generating graphics... => 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) generating graphics... => 4
([(0,1),(2,3),(2,4),(3,4)],5) generating graphics... => 2
([],6) generating graphics... => 0
([(4,5)],6) generating graphics... => 1
([(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(1,5),(2,5),(3,5),(4,5)],6) generating graphics... => 2
([(2,5),(3,4)],6) generating graphics... => 1
([(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) generating graphics... => 2
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 4
([(1,2),(3,4),(3,5),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) generating graphics... => 2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6) generating graphics... => 2
([(0,5),(1,4),(2,3)],6) generating graphics... => 1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) generating graphics... => 2
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) generating graphics... => 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) generating graphics... => 4
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) generating graphics... => 5
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6) generating graphics... => 3
([],7) generating graphics... => 0
([(5,6)],7) generating graphics... => 1
([(4,5),(4,6),(5,6)],7) generating graphics... => 2
([(2,6),(3,6),(4,6),(5,6)],7) generating graphics... => 2
([(3,6),(4,5)],7) generating graphics... => 1
([(3,5),(3,6),(4,5),(4,6)],7) generating graphics... => 2
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) generating graphics... => 2
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 4
([(2,3),(4,5),(4,6),(5,6)],7) generating graphics... => 2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7) generating graphics... => 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7) generating graphics... => 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7) generating graphics... => 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7) generating graphics... => 2
([(1,6),(2,6),(3,4),(3,5),(4,5)],7) generating graphics... => 2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7) generating graphics... => 2
([(1,6),(2,5),(3,4)],7) generating graphics... => 1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7) generating graphics... => 2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) generating graphics... => 2
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5)],7) generating graphics... => 3
([(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7) generating graphics... => 4
([(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 5
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) generating graphics... => 3
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7) generating graphics... => 3
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) generating graphics... => 2
([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7) generating graphics... => 2
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,5),(4,6)],7) generating graphics... => 4
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7) generating graphics... => 4
([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 5
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 6
([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) generating graphics... => 3
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) generating graphics... => 2
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 4
([(0,6),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) generating graphics... => 3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7) generating graphics... => 2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
([(0,3),(1,2),(4,5),(4,6),(5,6)],7) generating graphics... => 2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7) generating graphics... => 2
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7) generating graphics... => 2
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
([(0,1),(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) generating graphics... => 3
([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) generating graphics... => 4
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) generating graphics... => 3
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Code
def statistic(G):
    e = max([e for (e,_,_) in G.eigenvectors()])
    if e in ZZ:
        return e
Created
Apr 04, 2016 at 11:40 by Martin Rubey
Updated
Mar 23, 2017 at 20:51 by Martin Rubey