

A238408


The (rounded) 2nd modified Zagreb index of the rooted tree with Matula number n (n >= 2).


1



1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 1, 2, 2, 3, 2
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OFFSET

2,8


COMMENTS

The 2nd modified Zagreb index of a graph is defined as the summation of 1/(d(u)*d(v)) over all edges uv of G, where d(w) denotes the degree of the vertex w.
The Matula number of a rooted tree can be defined in the following recursive manner: to the onevertex tree there corresponds the number 1; to a tree T with root degree 1 there corresponds the tth prime number, where t is the Matula 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 numbers of the m branches of T.


REFERENCES

S. Nikolic, G. Kovacevic, A. Milicevic and N. Trinajstic, The Zagreb indices 20 years after, Croatica Chemica Acta, 76, 2003, 113124.
A. Milicevic, S. Nikolic and N. Trinajstic, On reformulated Zagreb indices, Molecular Diversity, 8, 2004, 393399.
F. Goebel, On a 11 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141143.
I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131142.
I. Gutman and YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 23142322.


LINKS

Table of n, a(n) for n=2..100.
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv1111.4288.
Index entries for sequences related to MatulaGoebel numbers


FORMULA

There are recurrence relations that give the 2nd modified Zagreb index of an "elevated" rooted tree (attach a new vertex to the root which becomes the root of the new tree) and of the merge of two rooted trees (identify the two roots). They make use of the sequence of the degrees of the level1 vertices (denoted by DL in the Maple program) .
In the Maple program, F(n) gives the actual (not rounded) 2nd modified Zagreb index of the rooted tree with Matula number n. For example, F(987654321) = 1979/300; the corresponding tree is the 29vertex tree given in Fig. 2 of the Deutsch reference.


EXAMPLE

a(5)=1; indeed the rooted tree with Matula number 5 is the path PQRS (rooted at P). The edges PQ and RS have endpoints of degrees 1 and 2 and the edge QR has endpoints of degrees 2 and 2; consequently, the contributions of these 3 edges to the 2nd modified Zagreb index are 1/2, 1/2 and 1/4, respectively; the 2nd modified Zagreb index is 5/4 = 1.25.


MAPLE

f := proc (x, y) options operator, arrow: 1/(x*y) end proc: c := 1: with(numtheory): F := proc (n) local DL, r, s: DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: if n = 2 then c elif bigomega(n) = 1 then F(pi(n))(sum(f(DL(pi(n))[j], bigomega(pi(n))), j = 1 .. bigomega(pi(n))))+sum(f(DL(pi(n))[j], 1+bigomega(pi(n))), j = 1 .. bigomega(pi(n)))+f(1, 1+bigomega(pi(n))) else F(r(n))+F(s(n))(sum(f(DL(r(n))[j], bigomega(r(n))), j = 1 .. bigomega(r(n))))(sum(f(DL(s(n))[j], bigomega(s(n))), j = 1 .. bigomega(s(n))))+sum(f(DL(r(n))[j], bigomega(n)), j = 1 .. bigomega(r(n)))+sum(f(DL(s(n))[j], bigomega(n)), j = 1 .. bigomega(s(n))) end if end proc: a := proc (n) options operator, arrow: round(F(n)) end proc: seq(a(n), n = 2 .. 100);


CROSSREFS

Cf. A238409.
Sequence in context: A232439 A199840 A126067 * A048858 A246465 A172497
Adjacent sequences: A238405 A238406 A238407 * A238409 A238410 A238411


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Feb 26 2014


STATUS

approved



