

A202852


MatulaGoebel numbers of rooted trees with no perfect matching and such that 2 is an eigenvalue of the Laplacian matrix.


0



343, 908, 1029, 1421, 1813, 2270, 2724, 2891, 3087, 3209, 3412, 3773, 3859, 4263, 4459, 4618, 4753, 4948, 5439, 5537, 5675, 5887, 6548, 6810, 7399, 7511
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OFFSET

1,1


COMMENTS

It is known that 2 is an eigenvalue of the Laplacian of any tree with a perfect matching (see the Ming & Zhang reference, Theorem 2).
The MatulaGoebel number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the MatulaGoebel number of the tree obtained from T by deleting the edge emanating from the root; to a tree T with root degree m>=2 there corresponds the product of the MatulaGoebel numbers of the m branches of T.


REFERENCES

F. Goebel, On a 11correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YeongNan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
Guo Ji Ming and Tan Shang Wang, A relation between the matching number and Laplacian spectrum of a graph, Linear Algebra and its Appl., 325, 2001, 7174.
Yizheng Fan, On the eigenvalue two and matching number of a tree, Acta Math. Appl. Sinica, English Series, 20, 2004, 257262.


LINKS

Table of n, a(n) for n=1..26.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

Set {A193402(n), n>=1} minus set {A193405(n), n>=1}.


EXAMPLE

The numbers 343, 908, and 3209 are in the sequence; they are the rooted trees obtained from the tree of Fig. 2 in the Fan reference by taking the root at different vertices. The tree has no perfect matching because it has 2 leaves with the same parent. Its Laplacian matrix has characteristic polynomial x(x2)(x5)(x1)^3*(x^2  4x + 1)^2.


CROSSREFS

Cf. A193402, A193405
Sequence in context: A043395 A336354 A269346 * A117689 A178374 A179147
Adjacent sequences: A202849 A202850 A202851 * A202853 A202854 A202855


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Feb 13 2012


STATUS

approved



