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A238407 The (rounded) first geometric-arithmetic index of the rooted tree with Matula number n (n >= 2). 0
1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 3, 4, 5, 3, 5, 5, 5, 5, 4, 6, 5, 6, 4, 5, 6, 5, 4, 6, 5, 6, 5, 4, 4, 6, 5, 5, 6, 4, 6, 7, 6, 6, 5, 5, 7, 6, 6, 4, 6, 7, 5, 5, 6, 5, 6, 5, 6, 7, 4, 7, 7, 4, 5, 7, 7, 5, 6, 6, 5, 8, 5, 7, 7, 6, 6, 8, 6, 6, 6, 7, 6, 7, 6, 5, 7, 6, 7, 7, 7, 6, 5, 7, 6, 8, 7 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

The first geometric-arithmetic index of a graph is defined as the summation of 2*sqrt(d(u)*d(v))/(d(u)+d(v)) over all edges uv of G, where d(w) denotes the degree of the vertex w.

The Matula 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  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 numbers of the m branches of T.

REFERENCES

D. Vukicevic and B. Furtula, Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges,  J. Math. Chem., 46, 2009, 1369-1376.

G.H. Fath-Tabar, B. Furtula and I. Gutman, A new geometric-arithmetic index, J. Math. Chem., 47, 2010, 477-486.

K. Ch. Das, I. Gutman and B. Furtula, On the first geometric-arithmetic index of graphs, Discrete Appl. Math., 159, 2011, 2030-2037.

F. Goebel, On a 1-1 correspondence between rooted trees and natural numbers, J. Combin. Theory, B 29 (1980), 141-143.

I. Gutman and A. Ivic, On Matula numbers, Discrete Math., 150, 1996, 131-142.

I. Gutman and Y-N. 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 Review, 10, 1968, 273.

E. Deutsch, Rooted tree statistics from Matula numbers, Discrete Applied Math., 160, 2012, 2314-2322.

LINKS

Table of n, a(n) for n=2..100.

E. Deutsch, Rooted tree statistics from Matula numbers, arXiv1111.4288.

Index entries for sequences related to Matula-Goebel numbers

FORMULA

There are recurrence relations that give the first geometric-arithmetic index 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 level-1 vertices (denoted by DL in the Maple program).

In the Maple program, F(n) gives the actual (not rounded) first geometric-arithmetic index of the rooted tree with Matula number n. For example, F(987654321) = 3+8*sqrt(2)/3+7*sqrt(3)/2+4*sqrt(5)/3+6*sqrt(6)/5+10*sqrt(10)/7+sqrt(15)/2; the corresponding tree is the 29-vertex tree given in Fig. 2 of the Deutsch reference.

EXAMPLE

a(5)=3; indeed the rooted tree with Matula 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 first geometric-arithmetic index are 2*sqrt(2)/3, 2*sqrt(2)/3, and 1, respectively; the sum-connectivity index is 4*sqrt(2)/3 + 1 =2.8856.

MAPLE

f := proc (x, y) options operator, arrow: 2*sqrt(x*y)/(x+y) end proc: c := 1: with(numtheory): F := 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 c elif bigomega(n) = 1 then F(pi(n))-(sum(f(DL(pi(n))[j], bigomega(pi(n))), j = 1 .. bigomega(pi(n))))+sum(f(DL(pi(n))[j], 1+bigomega(pi(n))), j = 1 .. bigomega(pi(n)))+f(1, 1+bigomega(pi(n))) else F(r(n))+F(s(n))-(sum(f(DL(r(n))[j], bigomega(r(n))), j = 1 .. bigomega(r(n))))-(sum(f(DL(s(n))[j], bigomega(s(n))), j = 1 .. bigomega(s(n))))+sum(f(DL(r(n))[j], bigomega(n)), j = 1 .. bigomega(r(n)))+sum(f(DL(s(n))[j], bigomega(n)), j = 1 .. bigomega(s(n))) end if end proc: a := proc (n) options operator, arrow: round(F(n)) end proc: seq(a(n), n = 2 .. 100);

CROSSREFS

Sequence in context: A070939 A113473 A265370 * A196050 A122027 A112751

Adjacent sequences:  A238404 A238405 A238406 * A238408 A238409 A238410

KEYWORD

nonn

AUTHOR

Emeric Deutsch, Feb 26 2014

STATUS

approved

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Last modified January 18 13:09 EST 2019. Contains 319271 sequences. (Running on oeis4.)