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, 1, 1, 2, 11, 26, 122, 781

Offset

1,5

Links

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

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

Gregory J. Clark and Joshua Cooper, Applications of the Harary-Sachs Theorem for Hypergraphs, arXiv:2107.10781 [math.CO], 2021.

J. Cooper and 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