

A184156


The Wiener polarity index of the rooted tree with MatulaGoebel number n.


0



0, 0, 0, 0, 1, 1, 0, 0, 2, 2, 2, 2, 2, 2, 3, 0, 2, 4, 0, 3, 3, 3, 4, 3, 4, 4, 6, 4, 3, 5, 3, 0, 4, 3, 4, 6, 3, 3, 5, 4, 4, 6, 4, 4, 7, 6, 5, 4, 4, 6, 4, 6, 0, 9, 5, 6, 4, 5, 3, 7, 6, 4, 8, 0, 6, 6, 3, 4, 7, 7, 4, 8, 6, 6, 8, 6, 5, 8, 4, 5, 12, 5, 6, 9, 5, 6, 6, 5, 4, 10, 6, 8, 5, 7, 5, 5, 6, 8, 8, 8, 6, 6, 9, 8, 9, 4, 6, 12, 5, 7
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OFFSET

1,9


COMMENTS

The Wiener polarity index of a connected graph G is the number of unordered pairs {i,j} of vertices of G such that the distance between i and j is 3.
The MatulaGoebel number of a rooted tree is 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

H. Deng, H. Xiao and F. Tang, On the extremal Wiener polarity index of trees with a given diameter, MATCH, Commun. Math. Comput. Chem., 63, 2010, 257264.
W. Du, X. Li and Y. Shi, Algorithms and extremal problem on Wiener polarity index, MATCH, Commun. Math. Comput. Chem., 62, 2009, 235244.
F. Goebel, On a 11correspondence 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 YeongNan 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.


LINKS



FORMULA

a(n) is the coefficient of x^3 in the Wiener polynomial of the rooted tree with MatulaGoebel number n. The coefficients of these Wiener polynomials are given in A196059. The Maple program is based on the above.


EXAMPLE

a(7)=0 because the rooted tree with MatulaGoebel number 7 is the rooted tree Y with no pair of vertices at distance 3.
a(11) = 2 because the rooted tree with MatulaGoebel number 7 is a path on 5 vertices, say a, b, c, d, e, with each of the pairs {a,d} and {b,e} at distance 3.


MAPLE

with(numtheory): WP := proc (n) local r, s, R: r := proc (n) options operator, arrow: op(1, factorset(n)) end proc: s := proc (n) options operator, arrow: n/r(n) end proc: R := proc (n) if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(x*R(pi(n))+x)) else sort(expand(R(r(n))+R(s(n)))) end if end proc: if n = 1 then 0 elif bigomega(n) = 1 then sort(expand(WP(pi(n))+x*R(pi(n))+x)) else sort(expand(WP(r(n))+WP(s(n))+R(r(n))*R(s(n)))) end if end proc: a := proc (n) options operator, arrow: coeff(WP(n), x, 3) end proc: seq(a(n), n = 1 .. 110);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



