A235112 a(n) = the largest of the M-indices of the trees with n vertices.

1, 2, 3, 7, 16, 32, 64, 152, 361, 1273, 4489, 22177, 109561, 735151

Offset: 1

Author

Emeric Deutsch, Jan 03 2014

Keywords

• nonn
• more

Comments

We define the M-index of a tree T to be the smallest of the Matula numbers of the rooted trees isomorphic (as a tree) to T. Example. The path tree P[5] = ABCDE has M-index 9. Indeed, there are 3 rooted trees isomorphic to P[5]: rooted at A, B, and C, respectively. Their Matula numbers are 11, 10, and 9, respectively. Consequently, the M-index of P[5] is 9.

a(n) = largest (= last) entry in row n of A235111.

It is conjectured that for n>=7 one has a(n) = A235120(n-6).

These numbers can be useful, for example, in the following situation. We intend to identify all trees that have 10 vertices and satisfy a certain property. Instead of scanning all rooted trees with Matula numbers from A005517(10)=125 to A005518(10)=219613, we do the scanning only for Matula numbers between 125 and a(10)=1273.

Examples

			a(4)=7. Indeed, there are 2 trees with 4 vertices: the path P[4] and the star S[3] with 3 edges. There are two rooted trees isomorphic to P[4]; they have Matula numbers 5 and 6; so the M-index is 5. There are two rooted trees isomorphic to S[3]; they have Matula numbers 7 and 8; so the M-index is 7. Max(5,7) = 7.
		

Links

• E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011.
• E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Appl. Math., 160, 2012, 2314-2322.
• 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.
• I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, The multiplicative version of the Wiener index, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.
• I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, On the multiplicative Wiener index and its possible chemical applications, Monatshefte f. Chemie, 131, 2000, 421-427.
• D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Rev. 10 (1968) 273.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A235111, A005518.

Formula

a(n) = A235111(n,A000055(n)).

Extensions

a(13) from Emeric Deutsch, Feb 16 2014

a(14) from Emeric Deutsch, Mar 12 2014