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A320648 a(n) is the number of connected Veblen hypergraphs (i.e., k-uniform hypergraphs where the degree of each vertex is divisible by k) with n edges up to isomorphism. 0
0, 0, 1, 1, 2, 11, 26, 122, 781 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

LINKS

Table of n, a(n) for n=1..9.

Gregory Clark, Joshua Cooper, A Harary-Sachs Theorem for Hypergraphs, arXiv:1812.00468 [math.CO], 2018.

J. Cooper, A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 3268-3292.

EXAMPLE

The only 3-uniform Veblen hypergraph with 3 edges is the single edge with multiplicity 3, {(1,2,3)^3}.

The only 3-uniform Veblen hypergraph with 4 edges is the 4-uniform simplex (i.e., the tetrahedron) as shown in Cooper and Dutle.

There are two 3-uniform Veblen hypergraphs with 5 edges: the crown, {(1,2,3),(1,2,4),(1,2,5),(3,4,5)^2}, and the tight 5-cycle, {(1,2,3),(2,3,4),(3,4,5),(4,5,1),(5,1,2)}.

PROG

(Sage)

e = n

#Given a 3-uniform hypergraph H, returns true if H is 3-valent.

def is_3_valent(H = IncidenceStructure([[]])):

    return (Set([H.degree(i) % 3 for i in range(len(H.ground_set()))]) == Set([0]))

#Returns a list of all connected 3-uniform Veblen hypergraphs with exactly e edges, up to isomorphism.

def Veblen_3graphs(e=1):

    V = []

    for n in range(3, e+2): #might be able to give a better bound

        for H in hypergraphs.nauty(e, n, uniform =3, multiple_sets = True, vertex_min_degree = 3, set_min_size = 3, connected = True):

            if is_3_valent(IncidenceStructure(H)):

                V.append(IncidenceStructure(H))

    return V

len(Veblen_3graphs(e))

CROSSREFS

Sequence in context: A104085 A080663 A248118 * A141464 A218471 A139211

Adjacent sequences:  A320645 A320646 A320647 * A320649 A320650 A320651

KEYWORD

nonn,more

AUTHOR

Gregory J. Clark, Oct 18 2018

STATUS

approved

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Last modified January 21 11:08 EST 2020. Contains 331105 sequences. (Running on oeis4.)