Identifier
Values
[1,0] => [2,1] => [2,1] => ([(0,1)],2) => 1
[1,0,1,1,0,0] => [3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4) => 1
[1,1,0,0,1,0] => [2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4) => 1
[1,0,1,0,1,1,0,0] => [4,1,2,5,3] => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,0,1,1,1,0,0,0] => [3,1,4,5,2] => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,0,0,1,0,1,0] => [2,5,1,3,4] => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,0,1,1,0,0,0] => [4,3,1,5,2] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 1
[1,1,1,0,0,0,1,0] => [2,3,5,1,4] => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5) => 1
[1,1,1,0,0,1,0,0] => [2,5,4,1,3] => [2,5,4,1,3] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 1
[1,1,1,0,1,0,0,0] => [5,3,4,1,2] => [3,5,1,4,2] => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 1
[1,0,1,0,1,0,1,1,0,0] => [5,1,2,3,6,4] => [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,0,1,0,1,1,1,0,0,0] => [4,1,2,5,6,3] => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,1,0,0,1,1,0,0] => [3,1,5,2,6,4] => [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,0,1,1,0,1,1,0,0,0] => [5,1,4,2,6,3] => [5,1,6,2,4,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 1
[1,0,1,1,1,0,1,0,0,0] => [6,1,4,5,2,3] => [4,6,1,2,5,3] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 1
[1,0,1,1,1,1,0,0,0,0] => [3,1,4,5,6,2] => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,1,0,0,1,0,1,0,1,0] => [2,6,1,3,4,5] => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,1,0,0,1,1,0,0,1,0] => [2,4,1,6,3,5] => [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => 1
[1,1,0,0,1,1,0,1,0,0] => [2,6,1,5,3,4] => [2,6,3,5,1,4] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,0,0,1,1,0,0] => [5,3,1,2,6,4] => [5,1,3,6,4,2] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,0,1,1,0,0,0] => [5,4,1,2,6,3] => [5,1,2,6,4,3] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1
[1,1,0,1,1,1,0,0,0,0] => [4,3,1,5,6,2] => [4,1,5,6,3,2] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,0,0,1,0,1,0] => [2,3,6,1,4,5] => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,1,1,0,0,1,0,0,1,0] => [2,6,4,1,3,5] => [2,4,6,5,1,3] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => 1
[1,1,1,0,0,1,0,1,0,0] => [2,6,5,1,3,4] => [2,3,6,5,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,0,1,0,0,0,1,0] => [6,3,4,1,2,5] => [3,4,6,1,5,2] => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 1
[1,1,1,0,1,1,0,0,0,0] => [5,3,4,1,6,2] => [3,1,5,4,6,2] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,1,0,0,0,0,1,0] => [2,3,4,6,1,5] => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 1
[1,1,1,1,0,0,0,1,0,0] => [2,3,6,5,1,4] => [2,6,5,1,3,4] => ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,1,1,1,0,0,1,0,0,0] => [2,6,4,5,1,3] => [2,6,1,4,3,5] => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 1
[1,1,1,1,0,1,0,0,0,0] => [6,3,4,5,1,2] => [3,6,1,4,5,2] => ([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 1
[1,0,1,0,1,0,1,0,1,1,0,0] => [6,1,2,3,4,7,5] => [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 1
[1,0,1,0,1,0,1,1,1,0,0,0] => [5,1,2,3,6,7,4] => [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,0,1,1,0,0,1,1,0,0] => [4,1,2,6,3,7,5] => [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => 1
[1,0,1,0,1,1,0,1,1,0,0,0] => [6,1,2,5,3,7,4] => [6,1,7,2,3,5,4] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 1
[1,0,1,0,1,1,1,0,1,0,0,0] => [7,1,2,5,6,3,4] => [5,7,1,2,3,6,4] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7) => 1
[1,0,1,0,1,1,1,1,0,0,0,0] => [4,1,2,5,6,7,3] => [4,1,2,5,6,7,3] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,0,1,1,0,0,1,1,1,0,0,0] => [3,1,5,2,6,7,4] => [3,1,5,6,2,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => 1
[1,0,1,1,0,1,0,0,1,1,0,0] => [6,1,4,2,3,7,5] => [6,1,2,7,4,5,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,1,0,1,0,1,1,0,0,0] => [6,1,5,2,3,7,4] => [6,1,2,7,3,5,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [5,1,4,2,6,7,3] => [5,1,2,6,4,7,3] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,0,0,1,1,0,0,0] => [3,1,6,5,2,7,4] => [3,1,6,5,2,7,4] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 1
[1,0,1,1,1,0,1,1,0,0,0,0] => [6,1,4,5,2,7,3] => [4,1,6,2,5,7,3] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [3,1,4,7,6,2,5] => [3,7,4,1,2,6,5] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,1,0,0,1,0,0,0] => [3,1,7,5,6,2,4] => [3,7,1,5,2,6,4] => ([(0,1),(0,6),(1,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,0,1,1,1,1,0,1,0,0,0,0] => [7,1,4,5,6,2,3] => [4,7,1,2,5,6,3] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => 1
[1,0,1,1,1,1,1,0,0,0,0,0] => [3,1,4,5,6,7,2] => [3,1,4,5,6,7,2] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 1
[1,1,0,0,1,0,1,0,1,0,1,0] => [2,7,1,3,4,5,6] => [2,3,4,5,7,1,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 1
[1,1,0,0,1,0,1,1,0,1,0,0] => [2,7,1,3,6,4,5] => [2,7,3,4,6,1,5] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,0,1,1,0,0,1,0,1,0] => [2,4,1,7,3,5,6] => [2,4,5,1,7,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => 1
[1,1,0,0,1,1,0,1,0,0,1,0] => [2,7,1,5,3,4,6] => [2,3,7,5,6,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,0,1,1,0,1,0,1,0,0] => [2,7,1,6,3,4,5] => [2,3,7,4,6,1,5] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,0,1,1,1,0,0,1,0,0] => [2,4,1,7,6,3,5] => [2,7,4,1,6,3,5] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,0,0,1,1,1,0,1,0,0,0] => [2,7,1,5,6,3,4] => [2,7,1,5,6,3,4] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,0,1,0,0,1,0,1,1,0,0] => [6,3,1,2,4,7,5] => [6,1,3,4,7,5,2] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,1,0,0,1,1,1,0,0,0] => [5,3,1,2,6,7,4] => [5,1,3,6,7,4,2] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,1,0,1,0,0,1,1,0,0] => [6,4,1,2,3,7,5] => [6,1,2,4,7,5,3] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,1,0,1,0,1,1,0,0,0] => [5,6,1,2,3,7,4] => [5,1,2,6,7,3,4] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [5,4,1,2,6,7,3] => [5,1,2,6,7,4,3] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,1,1,0,0,0,1,1,0,0] => [4,3,1,6,2,7,5] => [4,1,6,3,7,5,2] => ([(0,5),(1,4),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,0,1,1,0,0,1,1,0,0,0] => [6,3,1,5,2,7,4] => [3,1,6,2,7,5,4] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,1,1,0,1,1,0,0,0,0] => [6,4,1,5,2,7,3] => [4,1,6,2,7,5,3] => ([(0,5),(1,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [4,3,1,5,7,2,6] => [4,5,1,3,7,6,2] => ([(0,1),(0,6),(1,6),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [7,3,1,5,6,2,4] => [3,5,1,2,7,6,4] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [7,4,1,5,6,2,3] => [4,5,1,2,7,6,3] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [4,3,1,5,6,7,2] => [4,1,5,6,7,3,2] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,0,0,0,1,0,1,0,1,0] => [2,3,7,1,4,5,6] => [2,3,4,7,1,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,1,1,0,0,0,1,1,0,0,1,0] => [2,3,5,1,7,4,6] => [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7) => 1
[1,1,1,0,0,0,1,1,0,1,0,0] => [2,3,7,1,6,4,5] => [2,7,3,6,1,4,5] => ([(0,5),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,0,0,1,0,0,1,0,1,0] => [2,7,4,1,3,5,6] => [2,4,5,7,6,1,3] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 1
[1,1,1,0,0,1,0,1,0,0,1,0] => [2,7,5,1,3,4,6] => [2,3,5,7,6,1,4] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 1
[1,1,1,0,0,1,0,1,0,1,0,0] => [2,6,7,1,3,4,5] => [2,3,6,7,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,0,0,1,1,0,0,0,1,0] => [2,5,4,1,7,3,6] => [2,5,1,7,4,3,6] => ([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7) => 1
[1,1,1,0,0,1,1,0,0,1,0,0] => [2,7,4,1,6,3,5] => [2,4,1,7,6,3,5] => ([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [2,7,5,1,6,3,4] => [2,5,1,7,6,3,4] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,0,1,0,0,0,1,0,1,0] => [7,3,4,1,2,5,6] => [3,4,5,7,1,6,2] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6)],7) => 1
[1,1,1,0,1,0,0,0,1,1,0,0] => [6,3,4,1,2,7,5] => [3,1,6,7,4,5,2] => ([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [6,3,5,1,2,7,4] => [3,1,6,7,2,5,4] => ([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,0,1,0,1,1,0,0,0,0] => [6,5,4,1,2,7,3] => [6,1,2,7,5,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,0,1,1,0,0,0,0,1,0] => [5,3,4,1,7,2,6] => [3,5,1,7,4,6,2] => ([(0,1),(0,6),(1,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,0,1,1,0,0,0,1,0,0] => [7,3,4,1,6,2,5] => [3,4,1,7,2,6,5] => ([(0,3),(0,4),(1,2),(1,5),(2,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,0,1,1,0,0,1,0,0,0] => [7,3,5,1,6,2,4] => [3,5,1,7,2,6,4] => ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,0,1,1,1,0,0,0,0,0] => [5,3,4,1,6,7,2] => [3,1,5,6,4,7,2] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,1,0,0,0,0,1,0,1,0] => [2,3,4,7,1,5,6] => [2,3,7,1,4,5,6] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7) => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [2,3,7,5,1,4,6] => [2,3,7,5,1,4,6] => ([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [2,3,7,6,1,4,5] => [2,3,7,6,1,4,5] => ([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,1,0,0,1,0,0,0,1,0] => [2,7,4,5,1,3,6] => [2,4,7,1,5,3,6] => ([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [2,7,4,6,1,3,5] => [2,4,7,1,6,3,5] => ([(0,5),(1,3),(1,5),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,1,0,0,1,0,1,0,0,0] => [2,7,6,5,1,3,4] => [2,3,7,6,5,1,4] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,1,0,1,0,0,0,0,1,0] => [7,3,4,5,1,2,6] => [3,4,7,1,5,6,2] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [7,3,4,6,1,2,5] => [3,4,7,1,2,6,5] => ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,1,0,1,1,0,0,0,0,0] => [6,3,4,5,1,7,2] => [3,1,6,4,5,7,2] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,1,1,0,0,0,0,0,1,0] => [2,3,4,5,7,1,6] => [2,7,1,3,4,5,6] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7) => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [2,3,4,7,6,1,5] => [2,7,6,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[1,1,1,1,1,0,0,0,1,0,0,0] => [2,3,7,5,6,1,4] => [2,7,1,5,3,4,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,1,1,0,0,1,0,0,0,0] => [2,7,4,5,6,1,3] => [2,7,1,4,5,3,6] => ([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,1,1,0,1,0,0,0,0,0] => [7,3,4,5,6,1,2] => [3,7,1,4,5,6,2] => ([(0,1),(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
[] => [1] => [1] => ([],1) => 1
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searching the database for statistics with the same generating function
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$ \left(\begin{array}{rrrr} 4 & -1 & -2 & -1 \\ -1 & 4 & -1 & -2 \\ -2 & -1 & 4 & -1 \\ -1 & -2 & -1 & 4 \end{array}\right). $$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Map
graph of inversions
Description
The graph of inversions of a permutation.
For a permutation of $\{1,\dots,n\}$, this is the graph with vertices $\{1,\dots,n\}$, where $(i,j)$ is an edge if and only if it is an inversion of the permutation.
Map
cactus evacuation
Description
The cactus evacuation of a permutation.
This is the involution obtained by applying evacuation to the recording tableau, while preserving the insertion tableau.
Map
Ringel
Description
The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.