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

1,2

Comments

A graph G = (V,E) is harmonious if there exists an injective function f_V : V -> {0,1,...,n-1} such that a bijection occurs in the function f_E : E -> {0,...,n-1} after the harmoniously induced edge labels, f_E(v_iv_j) = (f_V(v_i) +f_V(v_j))(mod n), are applied.

A329910 contains the same data for simple graphs.

Links

Table of n, a(n) for n=1..20.

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).

Example

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.

Prog

(SageMath 9.2)
nlist = []
for n in range(1, 162):
    if (n % 2) == 0:
        nlist.append(((n^2/4) + (n/2))^(n/2))
    else:
        nlist.append(ceil(n/2)^n)

Crossrefs

A329910 but with self-loops.

For n odd, A110654 to the n-th power.

Sequence in context: A222469 A052618 A055142 * A183908 A280119 A224032

Adjacent sequences: A340231 A340232 A340233 * A340235 A340236 A340237

Keyword

easy,nonn

Author

Patrick D. Cone, Jan 01 2021

Status

approved