Identifier
Values
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 0
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,2),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 0
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => [1,1,1] => ([(0,1),(0,2),(1,2)],3) => 0
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Description
The minimal number of edges to add to make a graph Hamiltonian.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
A graph is Hamiltonian if it contains a cycle as a subgraph, which contains all vertices.
Map
to threshold graph
Description
The threshold graph corresponding to the composition.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
A threshold graph is a graph that can be obtained from the empty graph by adding successively isolated and dominating vertices.
A threshold graph is uniquely determined by its degree sequence.
The Laplacian spectrum of a threshold graph is integral. Interpreting it as an integer partition, it is the conjugate of the partition given by its degree sequence.
Map
chromatic difference sequence
Description
The chromatic difference sequence of a graph.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Let $G$ be a simple graph with chromatic number $\kappa$. Let $\alpha_m$ be the maximum number of vertices in a $m$-colorable subgraph of $G$. Set $\delta_m=\alpha_m-\alpha_{m-1}$. The sequence $\delta_1,\delta_2,\dots\delta_\kappa$ is the chromatic difference sequence of $G$.
All entries of the chromatic difference sequence are positive: $\alpha_m > \alpha_{m-1}$ for $m < \kappa$, because we can assign any uncolored vertex of a partial coloring with $m-1$ colors the color $m$. Therefore, the chromatic difference sequence is a composition of the number of vertices of $G$ into $\kappa$ parts.
Map
core
Description
The core of a graph.
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
The core of a graph $G$ is the smallest graph $C$ such that there is a homomorphism from $G$ to $C$ and a homomorphism from $C$ to $G$.
Note that the core of a graph is not necessarily connected, see [2].
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