
LINKS

Table of n, a(n) for n=1..9.
Gregory Clark, Joshua Cooper, A HararySachs Theorem for Hypergraphs, arXiv:1812.00468 [math.CO], 2018.
J. Cooper, A. Dutle, Spectra of uniform hypergraphs, Linear Algebra Appl. 436 (2012) 32683292.


EXAMPLE

The only 3uniform Veblen hypergraph with 3 edges is the single edge with multiplicity 3, {(1,2,3)^3}.
The only 3uniform Veblen hypergraph with 4 edges is the 4uniform simplex (i.e., the tetrahedron) as shown in Cooper and Dutle.
There are two 3uniform Veblen hypergraphs with 5 edges: the crown, {(1,2,3),(1,2,4),(1,2,5),(3,4,5)^2}, and the tight 5cycle, {(1,2,3),(2,3,4),(3,4,5),(4,5,1),(5,1,2)}.


PROG

(Sage)
e = n
#Given a 3uniform hypergraph H, returns true if H is 3valent.
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 3uniform 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))
