A196054 The second Zagreb index of the rooted tree with Matula-Goebel number n.

0, 1, 4, 4, 8, 8, 9, 9, 12, 12, 12, 14, 14, 14, 16, 16, 14, 19, 16, 18, 18, 16, 19, 22, 20, 19, 24, 21, 18, 23, 16, 25, 20, 18, 22, 28, 22, 22, 23, 26, 19, 26, 21, 22, 28, 24, 23, 32, 24, 27, 22, 26, 25, 34, 24, 30, 26, 23, 18, 32, 28, 20, 31, 36, 27, 27, 22, 24, 28, 30, 26, 39, 26, 28, 32, 30, 26, 31, 22, 36, 40, 23, 24, 36, 26, 26, 27, 30, 32, 38

Offset: 1

Author

Emeric Deutsch, Sep 28 2011

Keywords

• nonn

Comments

The second Zagreb index of a simple connected graph g is the sum of the degree products d(i)d(j) over all edges ij of g.

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.

Examples

			a(7)=9 because the rooted tree with Matula-Goebel number 7 is the rooted tree Y (1*3+3*1+3*1=9).
a(2^m) = m^2 because the rooted tree with Matula-Goebel number 2^m is a star with m edges.
		

Links

• F. Goebel, On a 1-1-correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.
• Ivan Gutman and Kinkar C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem. 50, 2004, 83-92.
• 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.
• S. Nikolic, G. Kovacevic, A. Milicevic, and N. Trinajstic, The Zagreb indices 30 years after, Croatica Chemica Acta, 76, 2003, 113-124.
• Index entries for sequences related to Matula-Goebel numbers

Crossrefs

Cf. A196052, A196053.

Programs

Maple

with(numtheory): a := proc (n) local r, s, b: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: b := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then 1+bigomega(pi(n)) else b(r(n))+b(s(n)) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then a(pi(n))+b(pi(n))+bigomega(pi(n))+1 else a(r(n))+a(s(n))+b(r(n))*bigomega(s(n))+b(s(n))*bigomega(r(n)) end if end proc: seq(a(n), n = 1 .. 90);

Formula

a(1)=0; if n=p(t) (the t-th prime), then a(n) = a(t)+b(t)+G(t)+1; if n=rs (r,s>=2), then a(n)=a(r)+a(s)+b(r)G(s)+b(s)G(r); here b(m) is the sum of the degrees of the nodes at level 1 of the rooted tree having Matula-Goebel number m and G(m) is the number of prime factors of m, counted with multiplicities. The Maple program is based on this recursive formula.