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 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.
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
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(x-2)(x-5)(x-1)^3*(x^2 - 4x + 1)^2.
CROSSREFS
KEYWORD
nonn
AUTHOR
Emeric Deutsch, Feb 13 2012
STATUS
approved