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A196065 The 1st multiplicative Zagreb index of the rooted tree with Matula-Goebel number n. 2
0, 1, 4, 4, 16, 16, 9, 9, 64, 64, 64, 36, 36, 36, 256, 16, 36, 144, 16, 144, 144, 256, 144, 64, 1024, 144, 576, 81, 144, 576, 256, 25, 1024, 144, 576, 256, 64, 64, 576, 256, 144, 324, 81, 576, 2304, 576, 576, 100, 324, 2304, 576, 324, 25, 1024, 4096, 144, 256, 576, 144, 1024, 256, 1024, 1296, 36, 2304 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The 1st multiplicative Zagreb index of a connected graph is the product of the squared degrees of the vertices of the graph. Alternatively, it is the square of the Narumi-Katayama index.

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, Multiplicative Zagreb indices of trees, Bull. Soc. Math. Banja Luka, 18, 2011, 17-23.

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.

H. Narumi and M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ., 16, 1984, 209-214.

Z. Tomovic and I. Gutman, Narumi-Katayama index of phenylenes, J. Serb. Chem. Soc., 66(4), 2001, 243-247.

LINKS

Table of n, a(n) for n=1..65.

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011

Index entries for sequences related to Matula-Goebel numbers

FORMULA

a(1)=0; a(2)=1, if n = p(t) (the t-th prime, t>=2), then a(n)=a(t)*(1+G(t))^2/G(t)^2; if n=rs (r,s>=2), then a(n)=a(r)*a(s)*G(n)^2/[G(r)*G(s)]^2; G(m) denotes the number of prime divisors of m counted with multiplicities. The Maple program is based on this recursive formula.

a(n) = (A196063(n))^2.

EXAMPLE

a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*9*1*1=9).

a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.

MAPLE

with(numtheory): a := proc (n) local r, s: 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 = 1 then 0 elif n = 2 then 1 elif bigomega(n) = 1 then a(pi(n))*(1+bigomega(pi(n)))^2/bigomega(pi(n))^2 else a(r(n))*a(s(n))*bigomega(n)^2/(bigomega(r(n))^2*bigomega(s(n))^2) end if end proc: seq(a(n), n = 1 .. 65);

CROSSREFS

Cf. A196063, A196064.

Sequence in context: A250069 A075882 A125757 * A258722 A264039 A196064

Adjacent sequences:  A196062 A196063 A196064 * A196066 A196067 A196068

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 01 2011

STATUS

approved

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Last modified January 18 03:13 EST 2019. Contains 319260 sequences. (Running on oeis4.)