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The number of labeled graphs placeable by a given permutation

✍ Scribed by Hasunuma, Toru; Shibata, Yukio

Publisher
John Wiley and Sons
Year
1996
Tongue
English
Weight
370 KB
Volume
21
Category
Article
ISSN
0364-9024

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✦ Synopsis

Let S be a finite set and u a permutation on S. The permutation u* on the set of 2-subsets of S is naturally induced by u. Suppose G is a graph and V(G), €(G) are the vertex set, the edge set, respectively. Let V(G) = S. If €(G) and u*(€(G)), the image of €(G) by u*, have no common element, then G is said to be placeable by u . This notion is generalized as follows. If any two sets of {€(G), (u')*(f(G)), . . . , (u'-')*(€(G))} have no common element, then G is said to be I-placeable by (T.

In this paper, w e count the number of labeled graphs which are I-placeable by a given permutation.

At first, w e introduce the interspaced I-th Fibonacci and Lucas numbers. When I = 2 these numbers are the ordinary Fibonacci and Lucas numbers. It is known that the Fibonacci and Lucas numbers are rounded powers. We show that the interspaced I-th Fibonacci and Lucas numbers are also rounded powers when I = 3. Next, w e show that the number of labeled graphs which are I-placeable by a given permutation is a product of the interspaced I-th Lucas numbers. Finally, using a property of the generalized binomial series, w e count the number of labeled graphs of size k which are I-placeable by u .

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