OFFSET
2,3
COMMENTS
The imbalance of an edge uv in a graph is defined as |d(u) - d(v)|, where d(w) denotes the degree of the vertex w. For example, in the path graph on 7 vertices, each of the two end-edges has imbalance 1 and each of the other four edges has imbalance 0.
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.
Sum of entries in row n = A196050(n) (= number of edges).
Sum(k*T(n,k), k>=0) = A238413(n) (= irregularity = sum of imbalances of all edges).
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
E. Deutsch, Rooted tree statistics from Matula numbers, arXiv1111.4288.
FORMULA
There are recurrence relations that give the generating polynomial of the edges with respect their imbalance 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, IP(n) gives the above mentioned generating polynomial of the rooted tree with Matula number n. For example, IP(987654321) = 3 + 7x + 9x^2 + 5x^3 + 4x^4; the corresponding tree is the 28-edge tree given in Fig. 2 of the Deutsch reference. S(n) yields the coefficients of the polynomial IP(n).
EXAMPLE
S(5)=1, 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 imbalances of these 3 edges are 0, 1, 1, and IP(5) = 1 + 2x.
Triangle starts:
1;
0,2;
0,2;
1,2;
1,2;
0,0,3;
0,0,3.
MAPLE
f := proc (x, y) options operator, arrow: abs(x-y) end proc: with(numtheory): IP := 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 1 elif bigomega(n) = 1 then IP(pi(n))-add(x^f(DL(pi(n))[j], bigomega(pi(n))), j = 1 .. bigomega(pi(n)))+add(x^f(DL(pi(n))[j], 1+bigomega(pi(n))), j = 1 .. bigomega(pi(n)))+x^bigomega(pi(n)) else IP(r(n))+IP(s(n))-add(x^f(DL(r(n))[j], bigomega(r(n))), j = 1 .. bigomega(r(n)))-add(x^f(DL(s(n))[j], bigomega(s(n))), j = 1 .. bigomega(s(n)))+add(x^f(DL(r(n))[j], bigomega(n)), j = 1 .. bigomega(r(n)))+add(x^f(DL(s(n))[j], bigomega(n)), j = 1 .. bigomega(s(n))) end if end proc: S := proc (n) options operator, arrow: seq(coeff(IP(n), x, q), q = 0 .. degree(IP(n))) end proc: for q from 2 to 20 do S(q) end do: # yields sequence in triangular form
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Mar 14 2014
STATUS
approved