OFFSET
0,33
COMMENTS
Initial terms computed using 'Nauty and Traces' (see the link).
T(0,r) = 1 because the "nodeless" graph has zero (therefore in this case all) nodes of degree r (for any r).
T(1,0) = 1 because only the empty graph on one node is 0-regular on 1 node.
T(1,r) = 0, for r>0: there's only one node and loops aren't allowed.
T(2,r) = 1, for r>0 since the only edges that are allowed are between the only two nodes.
T(3,r) = parity of r, for r>0. There are no such graphs of odd degree and for an even degree the only multigraph satisfying that condition is the regular triangular multigraph.
T(n,0) = 0, for n>1 because graphs having more than a node of degree zero are disconnected.
T(n,1) = 0, for n>2 since any connected graph with more than two nodes must have a node of degree greater than two.
T(n,2) = 1, for n>1: the only graphs satisfying that condition are the cyclic graphs of order n.
This sequence may be derived from A333330 by inverse Euler transform. - Andrew Howroyd, Mar 15 2020
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..350
Brendan McKay and Adolfo Piperno, Nauty and Traces
FORMULA
Column r is the inverse Euler transform of column r of A333330. - Andrew Howroyd, Mar 15 2020
EXAMPLE
Square matrix T(n,r) begins:
========================================================
n\r | 0 1 2 3 4 5 6 7
----+---------------------------------------------------
0 | 1, 1, 1, 1, 1, 1, 1, 1, ...
1 | 1, 0, 0, 0, 0, 0, 0, 0, ...
2 | 0, 1, 1, 1, 1, 1, 1, 1, ...
3 | 0, 0, 1, 0, 1, 0, 1, 0, ...
4 | 0, 0, 1, 2, 3, 4, 6, 7, ...
5 | 0, 0, 1, 0, 6, 0, 15, 0, ...
6 | 0, 0, 1, 6, 19, 49, 120, 263, ...
7 | 0, 0, 1, 0, 50, 0, 933, 0, ...
8 | 0, 0, 1, 20, 204, 1689, 13303, 90614, ...
...
PROG
# This program will execute the "else echo" line if the graph is nontrivial (first three columns, first two rows or both row and column indices are odd)
(nauty/shell)
for ((i=0; i<16; i++)); do
n=0
r=${i}
while ((n<=i)); do
if( (((r==0)) && ((n==0)) ) || ( ((r==0)) && ((n==1)) ) || ( ((r==1)) && ((n==2)) ) || ( ((r==2)) && !((n==1)) ) ); then
echo 1
elif( ((n==0)) || ((n==1)) || ((r==0)) || ((r==1)) || (! ((${r}%2 == 0)) && ! ((${n}%2 == 0)) || ( ((r==2)) && ((n==1)) )) ); then
echo 0
else echo $(./geng -c -d1 ${n} -q | ./multig -m${r} -r${r} -u 2>&1 | cut -d ' ' -f 7 | grep -v '^$'); fi;
((n++))
((r--))
done
done
CROSSREFS
KEYWORD
AUTHOR
Natan Arie Consigli, Dec 17 2019
STATUS
approved