

A235123


The rounded atombond connectivity (ABC) of the rooted tree with MatulaGoebel number n (n >= 2).


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0, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 3, 3, 4, 3, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 5, 4, 4, 4, 5, 4, 5, 4, 4, 5, 4, 4, 5, 5, 5, 4, 5, 4, 5, 5, 5, 5, 4, 4, 5, 5, 4, 5, 5, 5, 5, 4, 5, 5, 5, 5, 6, 5, 5, 6, 5, 5, 5, 4, 6, 6, 4, 4, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 6, 5, 5, 6, 6, 5, 5, 5, 5, 6, 5, 5, 6, 4, 6, 5, 6, 5, 5, 6, 5, 6, 4, 5, 6, 6, 5, 5, 5, 6, 6, 4, 6, 5, 6, 5, 6, 6, 5, 6, 6, 5, 6, 5, 6, 6, 5, 6, 6, 6, 5, 6, 5, 5, 7, 6, 6, 6, 6, 6, 6
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OFFSET

2,4


COMMENTS

The ABCindex of a graph is defined as the summation of sqrt[(d(u) + d(v)  2)/d(u)d(v)] over all edges uv of G, where d(w) denotes the degree of the vertex w.
The MatulaGoebel number of a rooted tree can be 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

B. Furtula, A. Graovac and D. Vukicevic, Atombond connectivity index of trees, Discrete Appl. Math., 157, 2009, 28282835.
F. Goebel, On a 11 correspondence 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 YN. Yeh, Deducing properties of trees from their Matula numbers, Publ. Inst. Math., 53 (67), 1993, 1722.
I. Gutman, B. Furtula and M. Ivanovic, Notes on trees with minimal atombond connectivity index, Comm. Math. Comp. Chem. (MATCH), 67, 2012, 467482.
R. Xing, B. Zhou and Zh. Du, Further results on atombond connectivity index of trees, Discrete Appl. Math., 158, 2010, 15361545.
D. W. Matula, A natural rooted tree enumeration by prime factorization, SIAM Review, 10, 1968, 273.
E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 23142322.


LINKS



FORMULA

There are recurrence relations that give the ABCindex of an "elevated" rooted tree (attach a new vertex to the root which becomes the root of the new tree) and of the merge of two rooted trees (identify the two roots). They make use of the sequence of the degrees of the level1 vertices (denoted by DL in the Maple program).
In the Maple program, A(n) gives the actual (not rounded) ABCindex of the rooted tree with Matula number n. For example, A(987654321) = (73/10)sqrt(2) + (7/3)sqrt(6)+(8/5)sqrt(5) + (2/5)sqrt(10); the corresponding tree is the 29vertex tree given in Fig. 2 of the Deutsch reference.


EXAMPLE

a(5)=2; indeed the rooted tree with MatulaGoebel number 5 is the path PQRS (rooted at P). The edges PQ and RS have endpoints of degrees 1 and 2 and the edge QR has endpoints of degrees 2 and 2; consequently, the contributions of these 3 edges to the ABCindex are 1/sqrt(2), 1/sqrt(2), and 1/sqrt(2), respectively; the ABCindex is 3/sqrt(2) = 2.1213.


MAPLE

with(numtheory): A := proc (n) local DL, r, s: DL := proc (n) if n = 2 then [1] elif bigomega(n) = 1 then [1+bigomega(pi(n))] else [op(DL(op(1, factorset(n)))), op(DL(n/op(1, factorset(n))))] end if end proc: 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 = 2 then 0 elif bigomega(n) = 1 then A(pi(n))+sqrt(bigomega(pi(n))/(1+bigomega(pi(n))))+add(sqrt((DL(pi(n))[j]+bigomega(pi(n))1)/(DL(pi(n))[j]*(1+bigomega(pi(n))))), j = 1 .. nops(DL(pi(n))))add(sqrt((DL(pi(n))[j]+bigomega(pi(n))2)/(DL(pi(n))[j]*bigomega(pi(n)))), j = 1 .. nops(DL(pi(n)))) else A(r(n))+A(s(n))add(sqrt((DL(r(n))[j]+bigomega(r(n))2)/(DL(r(n))[j]*bigomega(r(n)))), j = 1 .. nops(DL(r(n))))add(sqrt((DL(s(n))[j]+bigomega(s(n))2)/(DL(s(n))[j]*bigomega(s(n)))), j = 1 .. nops(DL(s(n))))+add(sqrt((DL(r(n))[j]+bigomega(n)2)/(DL(r(n))[j]*bigomega(n))), j = 1 .. nops(DL(r(n))))+add(sqrt((DL(s(n))[j]+bigomega(n)2)/(DL(s(n))[j]*bigomega(n))), j = 1 .. nops(DL(s(n)))) end if end proc: seq(round(A(n)), n = 2 .. 200);


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



