login
A338110
Number of spanning trees in the join of the disjoint union of two complete graphs each on n vertices with the empty graph on n vertices.
0
1, 128, 139968, 536870912, 5000000000000, 92442129447518208, 2988151979474457198592, 154742504910672534362390528, 12044329605471552321957641846784, 1342177280000000000000000000000000000, 206097683218942123873399068932507659403264, 42281678783395138381516145098915043145456549888
OFFSET
1,2
COMMENTS
Equivalently, the graph can be described as the graph on 3*n vertices with labels 0..3*n-1 and with i and j adjacent iff A011655(i + j) = 1.
These graphs are cographs.
LINKS
H-Y. Ching, R. Florez, and A. Mukherjee, Families of Integral Cographs within a Triangular Arrays, arXiv:2009.02770 [math.CO], 2020.
Eric Weisstein's World of Mathematics, Spanning Tree
FORMULA
a(n) = n*(2*n)^(3*(n - 1)).
a(n) = A193131(n)/3.
EXAMPLE
The adjacency matrix of the graph associated with n = 2 is: (compare A204437)
[0, 1, 1, 0, 1, 1]
[1, 0, 0, 1, 1, 0]
[1, 0, 0, 1, 0, 1]
[0, 1, 1, 0, 1, 1]
[1, 1, 0, 1, 0, 0]
[1, 0, 1, 1, 0, 0]
a(2) = 128 because the graph has 128 spanning trees.
MATHEMATICA
Table[n (2 n)^(3 (n - 1)), {n, 1, 10}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Rigoberto Florez, Oct 10 2020
STATUS
approved