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Identifier

St001261: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

([(0,1)],2) => 2

([],3) => 1

([(1,2)],3) => 2

([(0,2),(1,2)],3) => 2

([(0,1),(0,2),(1,2)],3) => 2

([],4) => 1

([(2,3)],4) => 2

([(1,3),(2,3)],4) => 2

([(0,3),(1,3),(2,3)],4) => 2

([(0,3),(1,2)],4) => 3

([(0,3),(1,2),(2,3)],4) => 2

([(1,2),(1,3),(2,3)],4) => 2

([(0,3),(1,2),(1,3),(2,3)],4) => 2

([(0,2),(0,3),(1,2),(1,3)],4) => 2

([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 2

([],5) => 1

([(3,4)],5) => 2

([(2,4),(3,4)],5) => 2

([(1,4),(2,4),(3,4)],5) => 2

([(0,4),(1,4),(2,4),(3,4)],5) => 2

([(1,4),(2,3)],5) => 3

([(1,4),(2,3),(3,4)],5) => 2

([(0,1),(2,4),(3,4)],5) => 3

([(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,4),(2,3),(3,4)],5) => 2

([(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 2

([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,3),(2,3),(2,4)],5) => 3

([(0,1),(2,3),(2,4),(3,4)],5) => 3

([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 3

([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 3

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => 2

([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 2

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 2

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 2

([],6) => 1

([(4,5)],6) => 2

([(3,5),(4,5)],6) => 2

([(2,5),(3,5),(4,5)],6) => 2

([(1,5),(2,5),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => 2

([(2,5),(3,4)],6) => 3

([(2,5),(3,4),(4,5)],6) => 2

([(1,2),(3,5),(4,5)],6) => 3

([(3,4),(3,5),(4,5)],6) => 2

([(1,5),(2,5),(3,4),(4,5)],6) => 2

([(0,1),(2,5),(3,5),(4,5)],6) => 3

([(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => 2

([(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(2,4),(2,5),(3,4),(3,5)],6) => 2

([(0,5),(1,5),(2,4),(3,4)],6) => 3

([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => 3

([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 2

([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 2

([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 3

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 2

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,4),(2,3)],6) => 4

([(1,5),(2,4),(3,4),(3,5)],6) => 3

([(0,1),(2,5),(3,4),(4,5)],6) => 3

([(1,2),(3,4),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 3

([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => 3

([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 3

([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => 3

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 2

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 2

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 3

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Description

The Castelnuovo-Mumford regularity of a graph.

References

[1] Trung, T. N. Regularity, matchings and Cameron-Walker graphs arXiv:1809.05377

Code

singular.eval('LIB "mregular.lib";')

def graph_ideal(G):
    """
    sage: G = graphs.CycleGraph(5)
    sage: graph_ideal(G)

    Lemma 7(1) of https://arxiv.org/pdf/1809.05377.pdf
    
    sage: all(statistic(G) == 1+sum(statistic(H)-1 for H in G.connected_components_subgraphs()) for G in graphs(5))
    True

    Lemma 2 of https://arxiv.org/pdf/1809.05377.pdf

    sage: set([(G.complement().is_chordal(), statistic(G)) for G in graphs(5) if G.degree().count(0) == 0])
    {(False, 3), (True, 2)}

    sage: set([(G.complement().is_chordal(), statistic(G)) for G in graphs(6) if G.degree().count(0) == 0])
    {(False, 3), (False, 4), (True, 2)}

    sage: set([(G.complement().is_chordal(), statistic(G)) for G in graphs(7) if G.degree().count(0) == 0])
    {(False, 3), (False, 4), (True, 2)}

    """
    G.relabel(inplace=False)
    R = PolynomialRing(QQ,['x_'+str(i) for i in G.vertices()])
    L = [R('x_'+str(i)) * R('x_'+str(j)) for i,j in G.edges(labels=False)]
    return R.ideal(L)

def statistic(G):
    if G.num_verts() == 0:
        return -1
    if G.num_edges() == 0:
        return 1
    return singular(graph_ideal(G)).regIdeal().sage()

Created

Sep 17, 2018 at 09:08 by Martin Rubey

Updated

Mar 05, 2020 at 15:49 by Martin Rubey