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Identifier
Values
([(0,1)],2) => ([],1) => ([],1) => ([],1) => 1
([(1,2)],3) => ([],1) => ([],1) => ([],1) => 1
([(2,3)],4) => ([],1) => ([],1) => ([],1) => 1
([(0,3),(1,2)],4) => ([],2) => ([],1) => ([],1) => 1
([(3,4)],5) => ([],1) => ([],1) => ([],1) => 1
([(1,4),(2,3)],5) => ([],2) => ([],1) => ([],1) => 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(4,5)],6) => ([],1) => ([],1) => ([],1) => 1
([(2,5),(3,4)],6) => ([],2) => ([],1) => ([],1) => 1
([(0,5),(1,4),(2,3)],6) => ([],3) => ([],1) => ([],1) => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(5,6)],7) => ([],1) => ([],1) => ([],1) => 1
([(3,6),(4,5)],7) => ([],2) => ([],1) => ([],1) => 1
([(1,6),(2,5),(3,4)],7) => ([],3) => ([],1) => ([],1) => 1
([(2,5),(2,6),(3,4),(3,6),(4,5)],7) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7) => ([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 2
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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
(4121141221411214).
Its eigenvalues are 0,4,4,6, so the statistic is 1.
The path on four vertices has eigenvalues 0,4.7,6,9.2 and therefore also statistic 1.
The graphs with statistic n1, n2 and n3 have been characterised, see [1].
Map
complement
Description
The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
Map
line graph
Description
The line graph of a graph.
Let G be a graph with edge set E. Then its line graph is the graph with vertex set E, such that two vertices e and f are adjacent if and only if they are incident to a common vertex in G.
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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].