OFFSET

1,3

COMMENTS

The 2nd multiplicative Zagreb index of a connected graph is the product of the products deg(i)*deg(j) taken over all edges ij of the graph (deg(v) denotes the degree of the vertex v). Alternatively, it is equal to the product of deg(i)^{deg(i)} over all vertices i of the graph.

The Matula-Goebel number of a rooted tree is 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.

LINKS

E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288 [math.CO], 2011.

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, Bulletin of International Mathematical Virtual Institut ISSN 1840-4367, Vol. 1, 2011, 13-19.

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 Rev. 10 (1968) 273.

FORMULA

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

EXAMPLE

a(7)=27 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y ((1*3)*(3*1)*(3*1)=27).

a(2^m) = m^m 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)))^(1+bigomega(pi(n)))/bigomega(pi(n))^bigomega(pi(n)) else a(r(n))*a(s(n))*bigomega(n)^bigomega(n)/(bigomega(r(n))^bigomega(r(n))*bigomega(s(n))^bigomega(s(n))) end if end proc: seq(a(n), n = 1 .. 55);

CROSSREFS

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Oct 01 2011

STATUS

approved