

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

Table of n, a(n) for n=1..110.
E. Deutsch, Tree statistics from Matula numbers, arXiv preprint arXiv:1111.4288, 2011
Index entries for sequences related to MatulaGoebel numbers


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

Cf. A196059
Sequence in context: A109913 A197169 A048052 * A187785 A238277 A258757
Adjacent sequences: A184153 A184154 A184155 * A184157 A184158 A184159


KEYWORD

nonn


AUTHOR

Emeric Deutsch, Oct 12 2011


STATUS

approved



