A340234 Number of harmonious graphs with n edges and at most n vertices, allowing self-loops.
1, 2, 8, 36, 243, 1728, 16384, 160000, 1953125, 24300000, 362797056, 5489031744, 96889010407, 1727094849536, 35184372088832, 722204136308736, 16677181699666569, 387420489000000000, 10000000000000000000, 259374246010000000000
Offset: 1
Examples
For n=3, the a(3) = 8 solutions are represented by the following adjacency matrices:
0 1 2 0 1 2 0 1 2 0 1 2
0 [ 1 1 1 ] 0 [ 1 1 0 ] 0 [ 1 0 1 ] 0 [ 1 0 0 ]
1 [ 1 0 0 ] 1 [ 1 1 0 ] 1 [ 0 0 0 ] 1 [ 0 1 0 ]
2 [ 1 0 0 ] 2 [ 0 0 0 ] 2 [ 1 0 1 ] 2 [ 0 0 1 ]
0 1 2 0 1 2 0 1 2 0 1 2
0 [ 0 1 1 ] 0 [ 0 1 0 ] 0 [ 0 0 1 ] 0 [ 0 0 0 ]
1 [ 1 0 1 ] 1 [ 1 1 1 ] 1 [ 0 0 1 ] 1 [ 0 1 1 ]
2 [ 1 1 0 ] 2 [ 0 1 0 ] 2 [ 1 1 1 ] 2 [ 0 1 1 ]
Notice that the number of self-loops in each graph is equal to the sum of the main diagonal.
Links
- Joseph A. Gallian, A dynamic survey of graph labeling, Elec. J. Combin., (2014), #DS6.
- D. Tanna, Harmonious Labeling of Certain Graphs, International Journal of Advanced Engineering Research and Studies, 2 (2013), 46-48.
Formula
For n odd, a(n) = ceiling(n/2)^n; for n even, a(n) = ((n^2/4) + (n/2))^(n/2) (conjectured).
Comments