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A235112 a(n) = the largest of the M-indices of the trees with n vertices. 1


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

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

%C 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.

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

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

%C 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.

%H E. Deutsch, <a href="http://arxiv.org/abs/1111.4288">Tree statistics from Matula numbers</a>, arXiv preprint arXiv:1111.4288, 2011.

%H E. Deutsch, <a href="http://dx.doi.org/10.1016/j.dam.2012.05.012">Rooted tree statistics from Matula numbers</a>, Discrete Appl. Math., 160, 2012, 2314-2322.

%H F. Goebel, <a href="http://dx.doi.org/10.1016/0095-8956(80)90049-0">On a 1-1-correspondence between rooted trees and natural numbers</a>, J. Combin. Theory, B 29 (1980), 141-143.

%H I. Gutman and A. Ivic, <a href="http://dx.doi.org/10.1016/0012-365X(95)00182-V">On Matula numbers</a>, Discrete Math., 150, 1996, 131-142.

%H I. Gutman and Yeong-Nan Yeh, <a href="http://www.emis.de/journals/PIMB/067/3.html">Deducing properties of trees from their Matula numbers</a>, Publ. Inst. Math., 53 (67), 1993, 17-22.

%H I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, <a href="http://dx.doi.org/10.1021/ci990060s">The multiplicative version of the Wiener index</a>, J. Chem. Inf. Comput. Sci., 40, 2000, 113-116.

%H I. Gutman, W. Linert, I. Lukovits, and Z. Tomovic, <a href="http://dx.doi.org/10.1007/PL00010312">On the multiplicative Wiener index and its possible chemical applications</a>, Monatshefte f. Chemie, 131, 2000, 421-427.

%H D. W. Matula, <a href="http://www.jstor.org/stable/2027327">A natural rooted tree enumeration by prime factorization</a>, SIAM Rev. 10 (1968) 273.

%H <a href="/index/Mat#matula">Index entries for sequences related to Matula-Goebel numbers</a>

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

%e 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.

%Y Cf. A235111, A005518.

%K nonn,more

%O 1,2

%A _Emeric Deutsch_, Jan 03 2014

%E a(13) from _Emeric Deutsch_, Feb 16 2014

%E a(14) from _Emeric Deutsch_, Mar 12 2014

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Last modified March 18 19:58 EDT 2019. Contains 321293 sequences. (Running on oeis4.)