

A238412


The multiplicative sum Zagreb index of the rooted tree with Matula number n (n >= 2).


0



2, 9, 9, 36, 36, 64, 64, 144, 144, 144, 240, 240, 240, 576, 625, 240, 900, 625, 960, 960, 576, 900, 2250, 2304, 900, 3375, 1536, 960, 3600, 576, 7776, 2304, 960, 3840, 8100, 2250, 2250, 3600, 9000, 900, 5760, 1536, 3840, 13500, 3375, 3600, 27216, 6400, 14400, 3840, 5760, 7776, 29160, 9216, 14000, 9000
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OFFSET

2,1


COMMENTS

The multiplicative sum Zagreb index of a graph is defined as the product of d(u) + d(v) over all edges uv of G, where d(w) denotes the degree of the vertex w.
The MatulaGoebel 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 MatulaGoebel 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 MatulaGoebel numbers of the m branches of T.


REFERENCES

E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 23142322.
M. Eliasi, A. Iranmanesh and I. Gutman, Multiplicative versions of first Zagreb index, Comm. Math. Comp. Chem. (MATCH), 68, 2012, 217230.
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.
K. Xu and K. Ch. Das, Trees, unicyclic, and bicyclic graphs extremal with respect to multiplicative sum Zagreb index, Comm. Math. Comp. Chem. (MATCH), 68, 2012, 257272.


LINKS

Table of n, a(n) for n=2..57.
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 multiplicative sum 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 level 1 vertices (denoted by DL in the Maple program).


EXAMPLE

a(5)=36; 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 multiplicative sum Zagreb index are 3, 3, 4; 3*3*4 = 36. a(987654321) = 92501790267801600000; the corresponding tree is the 29vertex tree given in Fig. 2 of the Deutsch reference.


MAPLE

f := proc (x, y) options operator, arrow: x+y end proc; c := 2; with(numtheory): a := 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 a(pi(n))*(2+bigomega(pi(n)))*(product(f(DL(pi(n))[j], 1+bigomega(pi(n)))/f(DL(pi(n))[j], bigomega(pi(n))), j = 1 .. bigomega(pi(n)))) else a(r(n))*a(s(n))*(product(f(DL(n)[j], bigomega(n)), j = 1 .. bigomega(n)))/((product(f(DL(r(n))[j], bigomega(r(n))), j = 1 .. bigomega(r(n))))*(product(f(DL(s(n))[j], bigomega(s(n))), j = 1 .. bigomega(s(n))))) end if end proc: seq(a(n), n = 2 .. 70);


CROSSREFS

Sequence in context: A197394 A198942 A168333 * A242064 A109322 A000587
Adjacent sequences: A238409 A238410 A238411 * A238413 A238414 A238415


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Feb 28 2014


STATUS

approved



