A238413 The irregularity of the rooted tree with Matula number n (n.>=2).

0, 2, 2, 2, 2, 6, 6, 2, 2, 2, 6, 6, 6, 2, 12, 6, 6, 12, 6, 6, 2, 6, 12, 2, 6, 6, 8, 6, 6, 2, 20, 2, 6, 6, 12, 12, 12, 6, 12, 6, 8, 8, 6, 6, 6, 6, 20, 10, 6, 6, 8, 20, 12, 2, 14, 12, 6, 6, 12, 12, 2, 8, 30, 6, 6, 12, 10, 6, 8, 12, 20, 8, 12, 6, 14, 6, 8, 6, 20, 12, 6, 6, 14, 6, 8, 6, 12, 20, 12, 10, 8, 2, 6, 12

Offset

2,2

Comments

The irregularity of a graph is defined as |the summation of |d(u) - d(v)| over all edges uv of G, where d(w) denotes the degree of the vertex w.

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.

References

M. O. Albertson, The irregularity of a graph, Ars Comb., 46 (1997) 219-225.

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..95.

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 irregularity 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) .

Example

a(5)=2; 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 irregularity 1, 0, and 1, respectively; the irregularity is 1 + 0 + 1 = 2.

Maple

f:=proc (x, y) options operator, arrow: abs(x-y) end proc: 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 f(1, 1) elif bigomega(n) = 1 then a(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 a(r(n))+a(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: seq(a(n), n = 2 .. 120);

Crossrefs

Sequence in context: A175809 A061033 A075094 * A151704 A110023 A279466

Adjacent sequences: A238410 A238411 A238412 * A238414 A238415 A238416

Keyword

nonn

Author

Emeric Deutsch, Mar 03 2014

Status

approved