Abstract
A graceful graph on n vertices is said to be simple if each of its connected components has at most one cycle and the component containing the edge with end points labelled 1 and n has no cycle. Let sn and tn denote the numbers of simple graceful graphs and graceful trees on n vertices respectively. Then tn≦sn≦p(An-2) where p(An-2) is the permanent (plus determinant) of the (n-2)×(n-2) matrix An-2=(aij) defined by:

More specifically, let ci denote the number of simple graceful graphs on n vertices with i cycles (i=0,1,2,...,k, where k=[(n-2)/3]). Then we have

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Reference
I. Cahit, "Are all complete binary trees graceful?" Amer. Math. Monthly 83(1976), 35–37.
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© 1978 Springer-Verlag
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Chen, C.C. (1978). On the enumeration of certain graceful graphs. In: Holton, D.A., Seberry, J. (eds) Combinatorial Mathematics. Lecture Notes in Mathematics, vol 686. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062523
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DOI: https://doi.org/10.1007/BFb0062523
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Print ISBN: 978-3-540-08953-7
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