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Your data matches 10 different statistics following compositions of up to 3 maps.
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Matching statistic: St001232
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> 0
1 => [1] => [1,0]
=> [1,0]
=> 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 3
11001 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4
001100 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
011001 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
100110 => [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> 4
110011 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 5
0011001 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
0110011 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
1001100 => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> 5
1100110 => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> 6
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St001416
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St001416: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St001416: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 1 => 1 = 0 + 1
1 => [1] => 1 => 1 => 1 = 0 + 1
00 => [2] => 10 => 11 => 2 = 1 + 1
01 => [1,1] => 11 => 10 => 1 = 0 + 1
10 => [1,1] => 11 => 10 => 1 = 0 + 1
11 => [2] => 10 => 11 => 2 = 1 + 1
001 => [2,1] => 101 => 111 => 3 = 2 + 1
010 => [1,1,1] => 111 => 101 => 3 = 2 + 1
011 => [1,2] => 110 => 100 => 2 = 1 + 1
100 => [1,2] => 110 => 100 => 2 = 1 + 1
101 => [1,1,1] => 111 => 101 => 3 = 2 + 1
110 => [2,1] => 101 => 111 => 3 = 2 + 1
0011 => [2,2] => 1010 => 1111 => 4 = 3 + 1
0110 => [1,2,1] => 1101 => 1000 => 3 = 2 + 1
1001 => [1,2,1] => 1101 => 1000 => 3 = 2 + 1
1100 => [2,2] => 1010 => 1111 => 4 = 3 + 1
00110 => [2,2,1] => 10101 => 11111 => 5 = 4 + 1
01100 => [1,2,2] => 11010 => 10000 => 4 = 3 + 1
10011 => [1,2,2] => 11010 => 10000 => 4 = 3 + 1
11001 => [2,2,1] => 10101 => 11111 => 5 = 4 + 1
001100 => [2,2,2] => 101010 => 111111 => 6 = 5 + 1
011001 => [1,2,2,1] => 110101 => 100000 => 5 = 4 + 1
100110 => [1,2,2,1] => 110101 => 100000 => 5 = 4 + 1
110011 => [2,2,2] => 101010 => 111111 => 6 = 5 + 1
0011001 => [2,2,2,1] => 1010101 => 1111111 => 7 = 6 + 1
0110011 => [1,2,2,2] => 1101010 => 1000000 => 6 = 5 + 1
1001100 => [1,2,2,2] => 1101010 => 1000000 => 6 = 5 + 1
1100110 => [2,2,2,1] => 1010101 => 1111111 => 7 = 6 + 1
Description
The length of a longest palindromic factor of a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the maximal length of a palindromic factor.
Matching statistic: St001417
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
St001417: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 1 => 1 = 0 + 1
1 => [1] => 1 => 1 => 1 = 0 + 1
00 => [2] => 10 => 11 => 2 = 1 + 1
01 => [1,1] => 11 => 10 => 1 = 0 + 1
10 => [1,1] => 11 => 10 => 1 = 0 + 1
11 => [2] => 10 => 11 => 2 = 1 + 1
001 => [2,1] => 101 => 111 => 3 = 2 + 1
010 => [1,1,1] => 111 => 101 => 3 = 2 + 1
011 => [1,2] => 110 => 100 => 2 = 1 + 1
100 => [1,2] => 110 => 100 => 2 = 1 + 1
101 => [1,1,1] => 111 => 101 => 3 = 2 + 1
110 => [2,1] => 101 => 111 => 3 = 2 + 1
0011 => [2,2] => 1010 => 1111 => 4 = 3 + 1
0110 => [1,2,1] => 1101 => 1000 => 3 = 2 + 1
1001 => [1,2,1] => 1101 => 1000 => 3 = 2 + 1
1100 => [2,2] => 1010 => 1111 => 4 = 3 + 1
00110 => [2,2,1] => 10101 => 11111 => 5 = 4 + 1
01100 => [1,2,2] => 11010 => 10000 => 4 = 3 + 1
10011 => [1,2,2] => 11010 => 10000 => 4 = 3 + 1
11001 => [2,2,1] => 10101 => 11111 => 5 = 4 + 1
001100 => [2,2,2] => 101010 => 111111 => 6 = 5 + 1
011001 => [1,2,2,1] => 110101 => 100000 => 5 = 4 + 1
100110 => [1,2,2,1] => 110101 => 100000 => 5 = 4 + 1
110011 => [2,2,2] => 101010 => 111111 => 6 = 5 + 1
0011001 => [2,2,2,1] => 1010101 => 1111111 => 7 = 6 + 1
0110011 => [1,2,2,2] => 1101010 => 1000000 => 6 = 5 + 1
1001100 => [1,2,2,2] => 1101010 => 1000000 => 6 = 5 + 1
1100110 => [2,2,2,1] => 1010101 => 1111111 => 7 = 6 + 1
Description
The length of a longest palindromic subword of a binary word.
A subword of a word is a word obtained by deleting letters. This statistic records the maximal length of a palindromic subword.
Any binary word of length $n$ contains a palindromic subword of length at least $n/2$, obtained by removing all occurrences of the letter which occurs less often. This bound is obtained by the word beginning with $n/2$ zeros and ending with $n/2$ ones.
Matching statistic: St001118
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 86%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 79% ●values known / values provided: 79%●distinct values known / distinct values provided: 86%
Values
0 => [1] => [1] => ([],1)
=> ? = 0
1 => [1] => [1] => ([],1)
=> ? = 0
00 => [2] => [1,1] => ([(0,1)],2)
=> 1
01 => [1,1] => [2] => ([],2)
=> ? = 0
10 => [1,1] => [2] => ([],2)
=> ? = 0
11 => [2] => [1,1] => ([(0,1)],2)
=> 1
001 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
010 => [1,1,1] => [3] => ([],3)
=> ? = 2
011 => [1,2] => [1,2] => ([(1,2)],3)
=> 1
100 => [1,2] => [1,2] => ([(1,2)],3)
=> 1
101 => [1,1,1] => [3] => ([],3)
=> ? = 2
110 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
0011 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
0110 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
1001 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> 2
1100 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3
00110 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
01100 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
10011 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> 3
11001 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4
001100 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
011001 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
100110 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
110011 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
0011001 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
0110011 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
1001100 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
1100110 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St000777
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
=> 1 = 0 + 1
1 => [1] => [1] => ([],1)
=> 1 = 0 + 1
00 => [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
01 => [1,1] => [2] => ([],2)
=> ? = 0 + 1
10 => [1,1] => [2] => ([],2)
=> ? = 0 + 1
11 => [2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
001 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
010 => [1,1,1] => [3] => ([],3)
=> ? = 2 + 1
011 => [1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
100 => [1,2] => [1,2] => ([(1,2)],3)
=> ? = 1 + 1
101 => [1,1,1] => [3] => ([],3)
=> ? = 2 + 1
110 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
0011 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
0110 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
1001 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 + 1
1100 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
00110 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
01100 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
10011 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 + 1
11001 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
001100 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
011001 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
100110 => [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 + 1
110011 => [2,2,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
0011001 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
0110011 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
1001100 => [1,2,2,2] => [1,2,2,2] => ([(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 5 + 1
1100110 => [2,2,2,1] => [2,2,2,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001207
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 29%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 29%
Values
0 => [2] => [1,1,0,0]
=> [2,3,1] => 2 = 0 + 2
1 => [1,1] => [1,0,1,0]
=> [3,1,2] => 2 = 0 + 2
00 => [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 1 + 2
01 => [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 2 = 0 + 2
10 => [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 2 = 0 + 2
11 => [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 1 + 2
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 2 + 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => ? = 1 + 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 1 + 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => ? = 2 + 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => ? = 2 + 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [2,3,6,1,4,5] => ? = 3 + 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => ? = 2 + 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [3,1,4,6,2,5] => ? = 2 + 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [4,1,2,5,6,3] => ? = 3 + 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,3,6,1,4,7,5] => ? = 4 + 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => ? = 3 + 2
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [3,1,4,7,2,5,6] => ? = 3 + 2
11001 => [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [4,1,2,5,7,3,6] => ? = 4 + 2
001100 => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [2,3,6,1,4,7,8,5] => ? = 5 + 2
011001 => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,5,1,3,6,8,4,7] => ? = 4 + 2
100110 => [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,1,4,7,2,5,8,6] => ? = 4 + 2
110011 => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [4,1,2,5,8,3,6,7] => ? = 5 + 2
0011001 => [3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [2,3,6,1,4,7,9,5,8] => ? = 6 + 2
0110011 => [2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [2,5,1,3,6,9,4,7,8] => ? = 5 + 2
1001100 => [1,3,1,3] => [1,0,1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> [3,1,4,7,2,5,8,9,6] => ? = 5 + 2
1100110 => [1,1,3,1,2] => [1,0,1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,1,2,5,8,3,6,9,7] => ? = 6 + 2
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
Matching statistic: St000741
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 0 + 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ? = 0 + 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ? = 2 + 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 3 + 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 2 + 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 2 + 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 3 + 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 4 + 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 3 + 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 3 + 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 4 + 1
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ? = 5 + 1
011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,5),(2,7),(2,10),(3,4),(3,6),(3,10),(4,11),(4,15),(5,12),(5,16),(6,8),(6,11),(7,9),(7,12),(8,14),(9,14),(10,15),(10,16),(11,13),(11,14),(12,13),(12,14),(13,15),(13,16)],17)
=> ? = 4 + 1
100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,5),(2,7),(2,10),(3,4),(3,6),(3,10),(4,11),(4,15),(5,12),(5,16),(6,8),(6,11),(7,9),(7,12),(8,14),(9,14),(10,15),(10,16),(11,13),(11,14),(12,13),(12,14),(13,15),(13,16)],17)
=> ? = 4 + 1
110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ? = 5 + 1
0011001 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 6 + 1
0110011 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 5 + 1
1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 5 + 1
1100110 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 6 + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St000454
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 29%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 29%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ? = 0 + 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 1 + 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 2 + 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ? = 2 + 1
0011 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 3 + 1
0110 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
1001 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
1100 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ? = 3 + 1
00110 => 01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 4 + 1
01100 => 00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 3 + 1
10011 => 11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 3 + 1
11001 => 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ? = 4 + 1
001100 => 011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,5),(2,7),(2,10),(3,4),(3,6),(3,10),(4,11),(4,15),(5,12),(5,16),(6,8),(6,11),(7,9),(7,12),(8,14),(9,14),(10,15),(10,16),(11,13),(11,14),(12,13),(12,14),(13,15),(13,16)],17)
=> ? = 5 + 1
011001 => 001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ? = 4 + 1
100110 => 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ? = 4 + 1
110011 => 100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,5),(2,7),(2,10),(3,4),(3,6),(3,10),(4,11),(4,15),(5,12),(5,16),(6,8),(6,11),(7,9),(7,12),(8,14),(9,14),(10,15),(10,16),(11,13),(11,14),(12,13),(12,14),(13,15),(13,16)],17)
=> ? = 5 + 1
0011001 => 0110011 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 6 + 1
0110011 => 0011001 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 5 + 1
1001100 => 1100110 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 5 + 1
1100110 => 1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ? = 6 + 1
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.
Matching statistic: St000455
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 - 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 - 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> -1 = 0 - 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> -1 = 0 - 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 0 = 1 - 1
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 2 - 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 1 - 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 1 - 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> ? = 2 - 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 2 - 1
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ([(0,4),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,7),(6,7),(7,8)],9)
=> ? = 2 - 1
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(0,5),(0,6),(1,7),(1,8),(2,5),(2,6),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8)],9)
=> ([(0,4),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,7),(6,7),(7,8)],9)
=> ? = 2 - 1
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,8),(5,8),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,8),(4,5),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 - 1
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 3 - 1
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 3 - 1
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(0,11),(0,12),(1,8),(1,12),(2,6),(2,7),(3,4),(3,11),(3,12),(4,5),(4,6),(5,9),(5,11),(6,9),(7,8),(7,9),(8,10),(9,10),(10,11),(10,12)],13)
=> ([(0,4),(0,5),(0,6),(0,7),(0,9),(0,10),(0,11),(0,12),(1,2),(1,3),(1,4),(1,6),(1,7),(1,8),(1,9),(1,12),(2,3),(2,6),(2,7),(2,8),(2,9),(2,11),(2,12),(3,5),(3,8),(3,9),(3,10),(3,11),(3,12),(4,5),(4,6),(4,7),(4,8),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,10),(5,11),(5,12),(6,7),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(8,9),(8,10),(8,12),(9,10),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 - 1
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ([(0,1),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,14),(0,15),(0,16),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,15),(1,16),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,16),(4,5),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,11),(8,12),(8,15),(8,16),(9,10),(9,11),(9,13),(9,15),(9,16),(10,12),(10,14),(10,15),(10,16),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 5 - 1
011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,5),(2,7),(2,10),(3,4),(3,6),(3,10),(4,11),(4,15),(5,12),(5,16),(6,8),(6,11),(7,9),(7,12),(8,14),(9,14),(10,15),(10,16),(11,13),(11,14),(12,13),(12,14),(13,15),(13,16)],17)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,3),(1,4),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,16),(2,3),(2,4),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(4,14),(4,15),(4,16),(5,6),(5,7),(5,8),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(9,10),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,14),(10,15),(10,16),(11,12),(11,13),(11,15),(12,14),(12,15),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 4 - 1
100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,5),(2,7),(2,10),(3,4),(3,6),(3,10),(4,11),(4,15),(5,12),(5,16),(6,8),(6,11),(7,9),(7,12),(8,14),(9,14),(10,15),(10,16),(11,13),(11,14),(12,13),(12,14),(13,15),(13,16)],17)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,3),(1,4),(1,6),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,14),(1,16),(2,3),(2,4),(2,6),(2,7),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(3,4),(3,5),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,15),(3,16),(4,5),(4,8),(4,9),(4,10),(4,11),(4,12),(4,14),(4,15),(4,16),(5,6),(5,7),(5,8),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,10),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(9,10),(9,11),(9,12),(9,13),(9,15),(9,16),(10,11),(10,12),(10,14),(10,15),(10,16),(11,12),(11,13),(11,15),(12,14),(12,15),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 4 - 1
110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(0,15),(0,16),(1,8),(1,9),(2,3),(2,4),(2,16),(3,6),(3,11),(4,5),(4,10),(5,8),(5,12),(6,9),(6,13),(7,12),(7,13),(7,15),(8,14),(9,14),(10,12),(10,15),(10,16),(11,13),(11,15),(11,16),(12,14),(13,14)],17)
=> ([(0,1),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(0,11),(0,12),(0,14),(0,15),(0,16),(1,4),(1,6),(1,7),(1,8),(1,9),(1,10),(1,11),(1,12),(1,13),(1,15),(1,16),(2,3),(2,4),(2,5),(2,6),(2,8),(2,9),(2,10),(2,11),(2,12),(2,13),(2,14),(2,16),(3,4),(3,5),(3,6),(3,7),(3,9),(3,10),(3,11),(3,12),(3,13),(3,14),(3,16),(4,5),(4,8),(4,9),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,8),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,7),(6,8),(6,9),(6,10),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,11),(8,12),(8,15),(8,16),(9,10),(9,11),(9,13),(9,15),(9,16),(10,12),(10,14),(10,15),(10,16),(11,12),(11,13),(11,14),(11,15),(12,13),(12,14),(12,15),(13,14),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 5 - 1
0011001 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 6 - 1
0110011 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 5 - 1
1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 5 - 1
1100110 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 6 - 1
Description
The second largest eigenvalue of a graph if it is integral.
This statistic is undefined if the second largest eigenvalue of the graph is not integral.
Chapter 4 of [1] provides lots of context.
Matching statistic: St001330
Values
0 => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
1 => ([(0,1)],2)
=> ([],2)
=> ([(0,1)],2)
=> 2 = 0 + 2
00 => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ? = 0 + 2
11 => ([(0,2),(2,1)],3)
=> ([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 3 = 1 + 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 1 + 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 2 + 2
0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 2
0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 2
1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,3),(0,4),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,5),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 2
1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 2
00110 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 2
01100 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 3 + 2
10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 3 + 2
11001 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ([(2,3),(2,10),(3,12),(4,5),(4,8),(4,12),(5,9),(5,11),(6,8),(6,9),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ([(0,4),(0,8),(0,10),(0,11),(0,12),(1,3),(1,7),(1,9),(1,10),(1,11),(1,12),(2,3),(2,5),(2,7),(2,8),(2,9),(2,11),(2,12),(3,7),(3,9),(3,10),(3,11),(3,12),(4,6),(4,8),(4,9),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(5,10),(5,11),(5,12),(6,7),(6,8),(6,9),(6,10),(6,11),(6,12),(7,9),(7,10),(7,11),(7,12),(8,9),(8,10),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,12)],13)
=> ? = 4 + 2
001100 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(2,3),(2,13),(3,14),(4,5),(4,12),(4,16),(5,11),(5,15),(6,11),(6,12),(6,14),(6,15),(6,16),(7,8),(7,10),(7,11),(7,13),(7,14),(7,15),(8,9),(8,12),(8,13),(8,14),(8,16),(9,10),(9,11),(9,14),(9,15),(9,16),(10,12),(10,14),(10,15),(10,16),(11,12),(11,16),(12,15),(13,14),(13,15),(13,16),(15,16)],17)
=> ([(0,2),(0,7),(0,9),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,6),(1,8),(1,11),(1,13),(1,14),(1,15),(1,16),(2,8),(2,9),(2,11),(2,12),(2,14),(2,15),(2,16),(3,5),(3,6),(3,8),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(4,5),(4,7),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,15),(5,16),(6,8),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,11),(8,13),(8,14),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 5 + 2
011001 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(2,3),(2,14),(3,13),(4,5),(4,11),(4,15),(5,12),(5,16),(6,10),(6,11),(6,12),(6,15),(6,16),(7,10),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,12),(8,13),(8,15),(8,16),(9,10),(9,11),(9,14),(9,15),(9,16),(10,13),(10,14),(11,12),(11,13),(11,16),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> ([(0,3),(0,7),(0,8),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,6),(1,8),(1,11),(1,13),(1,14),(1,15),(1,16),(2,4),(2,6),(2,10),(2,11),(2,12),(2,13),(2,15),(2,16),(3,5),(3,7),(3,10),(3,11),(3,12),(3,14),(3,15),(3,16),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,8),(6,9),(6,11),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,12),(7,13),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 4 + 2
100110 => ([(0,3),(0,4),(1,11),(1,16),(2,10),(2,15),(3,2),(3,13),(3,14),(4,1),(4,13),(4,14),(6,8),(7,9),(8,5),(9,5),(10,6),(11,7),(12,8),(12,9),(13,15),(13,16),(14,10),(14,11),(15,6),(15,12),(16,7),(16,12)],17)
=> ([(2,3),(2,14),(3,13),(4,5),(4,11),(4,15),(5,12),(5,16),(6,10),(6,11),(6,12),(6,15),(6,16),(7,10),(7,13),(7,14),(7,15),(7,16),(8,9),(8,10),(8,12),(8,13),(8,15),(8,16),(9,10),(9,11),(9,14),(9,15),(9,16),(10,13),(10,14),(11,12),(11,13),(11,16),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> ([(0,3),(0,7),(0,8),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,6),(1,8),(1,11),(1,13),(1,14),(1,15),(1,16),(2,4),(2,6),(2,10),(2,11),(2,12),(2,13),(2,15),(2,16),(3,5),(3,7),(3,10),(3,11),(3,12),(3,14),(3,15),(3,16),(4,6),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,7),(5,9),(5,10),(5,11),(5,12),(5,13),(5,14),(5,15),(5,16),(6,8),(6,9),(6,11),(6,13),(6,14),(6,15),(6,16),(7,8),(7,9),(7,12),(7,13),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,15),(8,16),(9,10),(9,11),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 4 + 2
110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ([(2,3),(2,13),(3,14),(4,5),(4,12),(4,16),(5,11),(5,15),(6,11),(6,12),(6,14),(6,15),(6,16),(7,8),(7,10),(7,11),(7,13),(7,14),(7,15),(8,9),(8,12),(8,13),(8,14),(8,16),(9,10),(9,11),(9,14),(9,15),(9,16),(10,12),(10,14),(10,15),(10,16),(11,12),(11,16),(12,15),(13,14),(13,15),(13,16),(15,16)],17)
=> ([(0,2),(0,7),(0,9),(0,12),(0,13),(0,14),(0,15),(0,16),(1,2),(1,6),(1,8),(1,11),(1,13),(1,14),(1,15),(1,16),(2,8),(2,9),(2,11),(2,12),(2,14),(2,15),(2,16),(3,5),(3,6),(3,8),(3,10),(3,11),(3,12),(3,13),(3,14),(3,15),(3,16),(4,5),(4,7),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(4,15),(4,16),(5,8),(5,9),(5,10),(5,11),(5,12),(5,13),(5,15),(5,16),(6,8),(6,10),(6,11),(6,12),(6,13),(6,14),(6,15),(6,16),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(7,15),(7,16),(8,11),(8,13),(8,14),(8,15),(8,16),(9,12),(9,13),(9,14),(9,15),(9,16),(10,11),(10,12),(10,13),(10,14),(10,15),(10,16),(11,13),(11,14),(11,15),(11,16),(12,13),(12,14),(12,15),(12,16),(13,15),(13,16),(14,15),(14,16),(15,16)],17)
=> ? = 5 + 2
0011001 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 6 + 2
0110011 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 5 + 2
1001100 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 5 + 2
1100110 => ([(0,3),(0,4),(1,18),(1,20),(2,17),(2,19),(3,1),(3,15),(3,16),(4,2),(4,15),(4,16),(6,8),(7,9),(8,10),(9,11),(10,5),(11,5),(12,10),(12,11),(13,8),(13,12),(14,9),(14,12),(15,19),(15,20),(16,17),(16,18),(17,13),(17,14),(18,6),(18,13),(19,7),(19,14),(20,6),(20,7)],21)
=> ?
=> ?
=> ? = 6 + 2
Description
The hat guessing number of a graph.
Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors.
Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
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