A connected labeled graph with 
graph edges in which all graph vertices can be labeled with distinct integers
(mod 
)
so that the sums of the pairs of numbers at the ends of
each graph edge are also distinct (mod 
). The ladder graph, fan,
wheel graph, Petersen
graph, tetrahedral graph, dodecahedral
graph, and icosahedral graph are all harmonious
(Graham and Sloane 1980).
Harmonious Graph
See also
Graceful Graph, Labeled Graph, Postage Stamp Problem, Sequential GraphExplore with Wolfram|Alpha
References
Gallian, J. A. "Open Problems in Grid Labeling." Amer. Math. Monthly 97, 133-135, 1990.Gardner, M. Wheels, Life, and other Mathematical Amusements. New York: W. H. Freeman, p. 164, 1983.Graham, R. L. and Sloane, N. "On Additive Bases and Harmonious Graphs." SIAM J. Algebraic Discrete Math. 1, 382-404, 1980.Guy, R. K. "The Corresponding Modular Covering Problem. Harmonious Labelling of Graphs." §C13 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 127-128, 1994.Referenced on Wolfram|Alpha
Harmonious GraphCite this as:
Weisstein, Eric W. "Harmonious Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HarmoniousGraph.html