A202852 Matula-Goebel numbers of rooted trees with no perfect matching and such that 2 is an eigenvalue of the Laplacian matrix.

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

Offset: 1

Author

Emeric Deutsch, Feb 13 2012

Keywords

• nonn

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 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.

Examples

			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.
		

References

• 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.

Links

• E. Deutsch, Rooted tree statistics from Matula numbers, arXiv:1111.4288.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A193402, A193405

Formula

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