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A202852
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Matula-Goebel numbers of rooted trees with no perfect matching and such that 2 is an eigenvalue of the Laplacian matrix.
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0
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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
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1,1
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COMMENTS
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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 Matula-Goebel number of a rooted tree can be defined in the following recursive manner: to the one-vertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the t-th prime number, where t is the Matula-Goebel 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 Matula-Goebel numbers of the m branches of T.
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REFERENCES
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F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.
I. Gutman and Yeong-Nan Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 17-22.
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, 71-74.
Yi-zheng Fan, On the eigenvalue two and matching number of a tree, Acta Math. Appl. Sinica, English Series, 20, 2004, 257-262.
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LINKS
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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 Matula-Goebel numbers
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FORMULA
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Set {A193402(n), n>=1} minus set {A193405(n), n>=1}.
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EXAMPLE
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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(x-2)(x-5)(x-1)^3*(x^2 - 4x + 1)^2.
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CROSSREFS
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Cf. A193402, A193405
Sequence in context: A043395 A336354 A269346 * A117689 A178374 A179147
Adjacent sequences: A202849 A202850 A202851 * A202853 A202854 A202855
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch, Feb 13 2012
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STATUS
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approved
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