ZBORNIK RADOVA Prirodno–matematičkog fakulteta
u Kragujevcu, 21 (1999) 111–119.
COLLECTION OF SCIENTIFIC PAPERS of the Faculty
of Science Kragujevac, 21 (1999) 111–119.
HOSOYA POLYNOMIALS OF TREES WITH UP TO 11 VERTICES
Dragan Stevanovića and Ivan Gutmanb
a
Faculty of Philosophy, Ćirila i Metodija 2, 18000 Niš, Yugoslavia, and
b
Faculty of Science, P. O. Box 60, 34000 Kragujevac, Yugoslavia
(Received March 7, 1999)
Abstract. Let G be a connected graph. If d(G, k) is the number of pairs of vertices of G that are at
P
d(G, k) xk is called the Hosoya polynomial of G . The Hosoya polynomials of
distance k apart, then
k≥0
all trees with 11 or less vertices (1 + 1 + 1 + 2 + 3 + 6 + 11 + 23 + 47 + 106 + 235 trees) are tabulated.
INTRODUCTION
The polynomial H(G, x) , which originally was called “Wiener polynomial ”, but which eventually
was re-named into “Hosoya polynomial ”, was invented by Hosoya in 1988 [1] as a counting polynomial
of potential interest in mathematical chemistry. The first works devoted to the study of its properties
appeared in 1993 and 1994 [2]–[4], whereas an extensive research along these lines started relatively
recently [5]–[9]. Most examinations of H(G, x), especially the recent ones [5, 6, 9], are concerned
with the case when G is a tree. In view of this, it would be advantageous to possess tables of the
Hosoya polynomials of trees. By means of them properties H(G, x) could be better envisaged, and
various conjectures could be checked (and counterexamples for them found). In the published, both
mathematical and chemical literature, such tables are not available. Our aim here is to fill this gap.
Let G be a connected graph of order n . Let d(G, k) be the number of pairs of vertices of G that
are distance k apart, k = 1, 2, . . . . In addition to this, d(G, 0) is equal to the number of vertices of
G , i. e., d(G, 0) = n . Then the Hosoya polynomial of G is defined as [1],
H(G) = H(G, x) =
D
X
k=0
d(G, k) xk
where D = D(G) is the diameter of the graph G , i. e., the greatest distance between two vertices
of G .
It should be noted that sometimes the Hosoya polynomial is defined in a slightly different manner,
P
namely as k≥1 d(G, k) xk . Even in this case the constant term d(G, 0) is fully determined: because
of the identity
D
X
k≥1
d(G, k) =
n
2
d(G, 0) can be calculated from d(G, 1), d(G, 2), . . . , d(G, D) .
The most remarkable property of the Hosoya polynomial is that its first derivative at x = 1 is
equal to the sum of the distances between all pairs of vertices of G , which is just the well–known
and much studied Wiener index, a graph invariant possessing numerous chemical applications. (This
fact was obviously the reason why Hosoya called H(G, x) “Wiener polynomial”.) Recently also the
higher derivatives of H(G, x) at x = 1 found applications in physical chemistry and pharmacology
[10].
THE TABLE
The table of Hosoya polynomials of trees is organized in the following manner. The trees are
grouped according to the number (n) of vertices, n = 1, 2, . . . , 11 . Within each group of trees with
a fixed number of vertices, the trees are ordered lexicographically with regard to the coefficients
d(G, k) , k = 0, 1, 2, ..., D(G) of the Hosoya polynomial; of course, for all such trees d(G, 0) = n and
d(G, 1) = n − 1 . The Hosoya polynomial is given below the picture of each tree.
1 tree with 1 vertex:
1 tree with 2 vertices:
r
r
1
1 tree with 3 vertices:
r
r
r
r
2
x +2x+3
x+2
2 trees with 4 vertices:
r
r
r
r
r
r
x3 +2x2 +3x+4
r
r
3x2 +3x+4
3 trees with 5 vertices:
r r
r
r
r
4
r
3
r
r
r
2
r
r
3
x +2x +3x +4x+5
r
r
2
r
r
2
2x +4x +4x+5
6x +4x+5
6 trees with 6 vertices:
r
r
r
r
r
r
r
r
x5 +2x4 +3x3 +4x2 +5x+6
r
r
r
r
r
r
r
r
r
r
4x3 +6x2 +5x+6
r
2x4 +3x3 +5x2 +5x+6
r
r
r
r
r
x4 +4x3 +5x2 +5x+6
r r r
r r
r
r
r
r
r
3x3 +7x2 +5x+6
r
r
10x2 +5x+6
11 trees with 7 vertices:
r
r
r
6
r
5
r
4
r
3
r
r
r
2
r
5
x +2x +3x +4x +5x +6x+7
r
4
r
3
r
r
2
2x +3x +4x +6x +6x+7
r
r
r
r
r
5
r
4
r
3
r
r
r
2
r
4
x +3x +5x +6x +6x+7
r
r
4
r
r
3
r
r
2
r
r
r
r
r
2
r
r
r
r
3
6x +9x +6x+7
r
2
r r
r
r
r
2
r
4
r
3
r
r
2
x +6x +8x +6x+7
r r r r
r r r
2
r
3
3x +6x +6x +6x+7
r
r
3
r
4
2x +6x +7x +6x+7
r r
r
3
r
4
3x +4x +8x +6x+7
r
r
3
r
4x +4x +7x +6x+7
r r
r
r
r
r
2
r
r
2
4x +11x +6x+7
15x +6x+7
23 trees with 8 vertices:
r
r
r
7
r
6
r
5
r
4
r
3
r
2
x +2x +3x +4x +5x +6x +7x+8
r
r
r
6
r
5
r
4
r
3
r
2
2x +3x +4x +5x +7x +7x+8
r
r
r
r
r
r
6
r
5
r
4
r
3
r
r
r
2
r
6
x +3x +4x +6x +7x +7x+8
r
5
r
4
r
3
r
r
2
x +2x +5x +6x +7x +7x+8
r
r
r
r
r
r
r
r r
r
r
r
4x5 +4x4 +5x3 +8x2 +7x+8
r
r
r
r
r
r
r
r
r
r
r
3x5 +4x4 +5x3 +9x2 +7x+8
r
r
r
r
r
2x5 +5x4 +6x3 +8x2 +7x+8
r
r
r
r
r
r
r
r
r
r
2x5 +5x4 +7x3 +7x2 +7x+8
r
r
2x5 +4x4 +7x3 +8x2 +7x+8
r
r
r
r
r
r
x5 +4x4 +8x3 +8x2 +7x+8
r r
r r
r
r
5
r
r
4
r
3
r
r
r
2
x +4x +7x +9x +7x+8
r r
r
4
r
r
r
3
r
r
2
r
r
4
6x +5x +10x +7x+8
3
r
r
2
5x +8x +8x +7x+8
r
r
r
r
r
4
r
3
r r r
r
r
r
r
2
r
4
r r
r
r
3
r
r
2
r
r
4
r
3
r
2
4x +8x +9x +7x+8
4x +5x +12x +7x+8
3x +9x +9x +7x+8
r
r r
r r r
r
r
4
r r
r
3
r
r
r
2
3x +8x +10x +7x+8
r
3
r
r
r
r
r
3
2
r
r
r
r
r
3
9x +12x +7x+8
r
r
3
r r r r r
r r r r
r
r
2
r
r
3
8x +13x +7x+8
r
2
x +8x +12x +7x+8
r r r r
2
r
4
2x +9x +10x +7x+8
r r r r
r
r
4
r
r
2
r
r
2
5x +16x +7x+8
21x +7x+8
47 trees with 9 vertices:
r
r
r
r
r
r
r
r
r
r
r
x8 +2x7 +3x6 +4x5 +5x4 +6x3 +7x2 +8x+9
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r r
r
4x6 +4x5 +5x4 +6x3 +9x2 +8x+9
r
r
r
r
r
r
r
3x6 +4x5 +5x4 +6x3 +10x2 +8x+9
r
r
r
r
r
6
r
5
r
4
r
r
3
r
2
2x +5x +5x +7x +9x +8x+9
r
r
x7 +2x6 +4x5 +6x4 +7x3 +8x2 +8x+9
r
r
r
r
r
x7 +3x6 +4x5 +5x4 +7x3 +8x2 +8x+9
r
r
2x7 +3x6 +4x5 +5x4 +6x3 +8x2 +8x+9
r
r
r
r
6
r
5
r
4
r
3
r
r
2
2x +4x +6x +8x +8x +8x+9
r
r
r
r
r
r
r
r
r
r
6
r
5
4
r
3
r
r
r
2
r
r
r
6
2x +4x +5x +8x +9x +8x+9
5
r
4
r
3
r
r
2
2x +3x +7x +7x +9x +8x+9
r
r
r
r
r
r
r
r
r
r
r
x6 +4x5 +7x4 +8x3 +8x2 +8x+9
r
r
r
r
r
r
r
r
r
x6 +4x5 +5x4 +8x3 +10x2 +8x+9
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
x6 +3x5 +6x4 +9x3 +9x2 +8x+9
r r
r
r
x6 +4x5 +6x4 +8x3 +9x2 +8x+9
r r
r
r
r
r
x6 +2x5 +7x4 +8x3 +10x2 +8x+9
r r
r
r
r
r
r
6x5 +5x4 +6x3 +11x2 +8x+9
r
r
r
r
r
r
5
r
r
4
3
r
r
r
2
r
5
4x +7x +8x +9x +8x+9
r
r
r
4
r
3
r r r
r
r
r
2
r
5
4x +6x +8x +10x +8x+9
r
4
r
3
r
r
2
4x +5x +6x +13x +8x+9
r r
r r
r
r
r
r
5
r
4
3
r
r
r
2
r
5
3x +7x +9x +9x +8x+9
r
r
4
r
3
r
r
r r
r
r
r
r
2
5
3x +7x +7x +11x +8x+9
4
r
3
r
r
2
3x +5x +9x +11x +8x+9
r
r
r
r
5
r
r
r
r
4
3
r
r
r
2
r
5
2x +7x +10x +9x +8x+9
r r
r
4
r
r
3
r
r
r
2
r
5
2x +7x +8x +11x +8x+9
r
r
r
r
4
3
r
r
2
2x +6x +10x +10x +8x+9
r
r r
r
r
r
r
r
r
r
2x5 +6x4 +10x3 +10x2 +8x+9
r
r r
r
r
r
r
r
r
2x5 +5x4 +10x3 +11x2 +8x+9
r r
r
r
r
r
r r
r
r
5
r
4
r
3
r
r
r
2
x +5x +9x +13x +8x+9
r
r
r
3
r
2
r
8x +6x +14x +8x+9
r
3
r
r
r
r
2
r
r
r r
r
r
2
r
7x +10x +11x +8x+9
r
r
r
2
8x +10x +10x +8x+9
r r
3
r
3
r r r
r
4
r
r
4
9x +6x +13x +8x+9
r r r
r
4
r r
r
4
r
r
4
r
x5 +5x4 +11x3 +11x2 +8x+9
r
r r r
r
r
3
r
2
r
6x +12x +10x +8x+9
r r
r
r r
r
r
r
r
r
r
r
r
6x4 +10x3 +12x2 +8x+9
r
r
r r r r
r
r
r
r
5x4 +12x3 +11x2 +8x+9
r
r
r
r
5x4 +6x3 +17x2 +8x+9
r
r
r
r r
r
r
r
4
r
3
r
r
2
r
r
4
4x +12x +12x +8x+9
r
r
4
r
3
r
r
2
r
r
r
r
4
r
r
2
r
r
r
12x +16x +8x+9
r
r
r
3
r
2
x +10x +17x +8x+9
r r r r r r
r r r r r
r
3
r
4
2x +12x +14x +8x+9
2
r
2
r r r r
r
3
r
3
3x +12x +13x +8x+9
r r r r r
r
r
4
r r r r
r r r r r
3
r
2
4x +10x +14x +8x+9
3x +12x +13x +8x+9
r
r
3
r r r r
r
r r r
r r r r
r
r
2
r
3
10x +18x +8x+9
r
r
r
r
2
r
2
6x +22x +8x+9
28x +8x+9
106 trees with 10 vertices:
r
r
r
9
r
8
r
7
r
6
r
5
r
4
r
3
r
r
r
2
r
8
x +2x +3x +4x +5x +6x +7x +8x +9x+10
r
7
r
8
r
6
r
5
r
4
r
3
r
r
r
2
r
8
x +3x +4x +5x +6x +8x +9x +9x+10
r
8
r
7
r
6
r
5
r
4
r
3
r
r
r
2
r
r
6
r
5
r
6
r
5
r
r
5
r
4
r
4
r
3
r
4
r
3
r
r
2
3x +4x +5x +6x +7x +11x +9x+10
r
2
r
3
r
2
r
r
r
7
r
6
r
5
r
4
r
3
r
r
2
2x +5x +5x +6x +8x +10x +9x+10
r
r
r
r
r
r
r
r
r
r
2x7 +4x6 +5x5 +7x4 +9x3 +9x2 +9x+10
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
2x7 +3x6 +5x5 +8x4 +8x3 +10x2 +9x+10
r r
r
r
r
2x7 +4x6 +5x5 +6x4 +9x3 +10x2 +9x+10
r
2x7 +3x6 +6x5 +7x4 +8x3 +10x2 +9x+10
r
r
r
r
r
r
x7 +4x6 +6x5 +6x4 +9x3 +10x2 +9x+10
r
4x +4x +5x +6x +7x +10x +9x+10
r
6
r
2
r
r
7
r r
7
r
7
r
x +2x +3x +6x +7x +8x +9x +9x+10
r
r
3
x +2x +4x +5x +7x +8x +9x +9x+10
r
r
r
4
r
r
7
r
5
r
r
2x +3x +4x +5x +6x +7x +9x +9x+10
r
r
r
6
r
r
r
r
r
r
r
x7 +4x6 +5x5 +6x4 +9x3 +11x2 +9x+10
r
r
r
r
r
r
r
r
r
r
r
r
r
x7 +3x6 +6x5 +8x4 +9x3 +9x2 +9x+10
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
x7 +2x6 +5x5 +8x4 +9x3 +11x2 +9x+10
r
r
r
r
r
r
r
r
5
r
4
r
r
3
r
r
2
4x +6x +7x +9x +10x +9x+10
r
r
6
r
r
6
r
r
5
r
4
r
r
5
r
4
r
3
r
r
2
4x +5x +6x +7x +14x +9x+10
r
r
3
r
2
4x +6x +6x +9x +11x +9x+10
r r r
r
r
r
r
6
r
5
r
r
4
r
r
3
r
2
4x +4x +9x +8x +11x +9x+10
r
r
r r
r
r
6
r
r
r
5
r
4
r
3
r
r
2
3x +7x +6x +8x +12x +9x+10
r
r
r
6
r
5
4
r
3
r
r
2
3x +6x +9x +9x +9x +9x+10
r r
r
r
r
r
r
r
r
r
3x6 +5x5 +8x4 +10x3 +10x2 +9x+10
r r
r
r
r
r r
r
r
r
r
r
r
r
r
r
r
3x6 +4x5 +9x4 +8x3 +12x2 +9x+10
r
r
r
r
r
r
r
5
r
r
3
r
r
2
2x +6x +9x +9x +10x +9x+10
r
r
r
r
6
r
r
6
r
4
r
5
r
4
r
r
3
r
r
2
r
2x +6x +7x +10x +11x +9x+10
r
3
r
r
2
2x +6x +8x +10x +10x +9x+10
r
r
5
r
r
r
4
r
2x6 +7x5 +6x4 +9x3 +12x2 +9x+10
r
r
6
r
r r
r
r
r
3x6 +5x5 +6x4 +10x3 +12x2 +9x+10
r
r
r
6x6 +5x5 +6x4 +7x3 +12x2 +9x+10
r
r
6
r
r r
r
r
r
r
r
x7 +2x6 +5x5 +8x4 +10x3 +10x2 +9x+10
r r
r
r
x7 +3x6 +5x5 +8x4 +9x3 +10x2 +9x+10
x7 +3x6 +5x5 +7x4 +10x3 +10x2 +9x+10
r
r
r
r
r
6
r
5
r
r
4
r
3
r
2
r
2x +5x +8x +10x +11x +9x+10
r
r r
r
r
r
r
r
r
r
r
2x6 +5x5 +7x4 +11x3 +11x2 +9x+10
r
r r
r
r
r
r
r
r
2x6 +5x5 +6x4 +11x3 +12x2 +9x+10
r
r
r
r
r
6
r
5
r
r
4
r
3
r
r
r
2
r
6
2x +4x +9x +11x +10x +9x+10
r
r
r
5
r
r
4
3
r
r
2
2x +4x +8x +11x +11x +9x+10
r r
r
r
r r
r
6
r
5
r
r
4
r
3
r
r
r
2
6
2x +3x +10x +9x +12x +9x+10
r
r
5
4
r
r
3
r
r
2
x +6x +9x +10x +10x +9x+10
r
r
6
r
r
r
r
r
5
4
r
r
3
r
r
r
2
6
x +5x +9x +11x +10x +9x+10
r
r
5
4
r r
r
r
3
r
r
2
x +5x +8x +10x +12x +9x+10
r
r r r
r
r
r
r r
r
r
r
r
r
x6 +5x5 +6x4 +10x3 +14x2 +9x+10
r
r
r
r
r
r
r
r
r
r
r
r r
r
r
r
r
r
r
r
x6 +4x5 +9x4 +11x3 +11x2 +9x+10
r
r
x6 +4x5 +8x4 +12x3 +11x2 +9x+10
r
r
r
r r
r
r
r
r
r
r
x6 +4x5 +7x4 +12x3 +12x2 +9x+10
r r r
r
r
r
r
x6 +3x5 +8x4 +12x3 +12x2 +9x+10
r
r
r
r
r
r
x6 +2x5 +9x4 +10x3 +14x2 +9x+10
r r
r r
r
r r
r
5
r
4
r
3
r
r
r
r
2
r
5
9x +6x +7x +14x +9x+10
r r r
r
4
r
3
r
r
r
r
2
r
5
8x +6x +7x +15x +9x+10
r
r
4
r
3
r
r
2
6x +9x +10x +11x +9x+10
r
r r
r
r
r
5
r
4
r
r
3
r
r
r
2
r
5
6x +9x +9x +12x +9x+10
r
r
4
r
5
r
4
3
r
r
2
5x +6x +7x +18x +9x+10
r
r
3
r
r
r
r
r
r
r
r
r
r
r r
r
r
r
r
4
r
r
3
3
r
6x +7x +10x +13x +9x+10
5
4
r
6x +8x +9x +13x +9x+10
r
5
r r
2
r r r r
r
r r
r
2
r
4x +10x +12x +10x +9x+10
r
5
2
r
r
r
4
r
3
r
2
r
4x +10x +11x +11x +9x+10
r
r
r
r
r
r
r
r
r
r
r
r
r
4x5 +9x4 +12x3 +11x2 +9x+10
r r r
r
r
r
r
r r
r
r
r
r
r
r
r
4x5 +9x4 +11x3 +12x2 +9x+10
r
r
r r r
r
r
4x5 +9x4 +8x3 +15x2 +9x+10
r
r
r
r
r
r
r
r
r
r
r
r
r
4x5 +9x4 +11x3 +12x2 +9x+10
r
r
r
4x5 +8x4 +12x3 +12x2 +9x+10
r
r
r r
r
r
r
r
4x5 +8x4 +11x3 +13x2 +9x+10
r r
r r r r
r
r
5
r
r
4
r
r
3
r
r
2
r
r
5
4x +6x +11x +15x +9x+10
r
r
r
2
r r
r
r
5
4
r
r
3
r
2
3x +10x +9x +14x +9x+10
r r
r
r
4
r
3
3x +10x +12x +11x +9x+10
r r
5
r r
r
4
r r
r
r
r
r
3
r
r
2
r
5
3x +9x +13x +11x +9x+10
r
r
r
4
r
r
3
r
r
2
r
5
3x +8x +13x +12x +9x+10
r
r
4
r r
r
r
3
r
2
3x +8x +12x +13x +9x+10
r
r r r r
r
r
5
r
4
r
r
3
r
r
2
r
r
r
r
5
3x +6x +13x +14x +9x+10
r r
4
r
r
3
r
r
2
r
5
2x +9x +13x +12x +9x+10
r r r
r
4
r
r
3
r
2
2x +9x +10x +15x +9x+10
r
r
r
r
r r
r
r
r
r
r
2x5 +8x4 +14x3 +12x2 +9x+10
r r
r
r
r
r
r
r
r
r
2x5 +8x4 +13x3 +13x2 +9x+10
r
r
r r
r
r
r
r
r
2x5 +7x4 +14x3 +13x2 +9x+10
r
r r r
r
r
5
r
4
r r r r
r r r r
r
3
r
r
r
2
r
5
2x +7x +13x +14x +9x+10
r
4
r
r
3
r
2
2x +6x +13x +15x +9x+10
r
r
5
4
r
r
3
r
r
2
x +6x +15x +14x +9x+10
r r
r r r r
r
r
5
4
r
r r r r
r
3
r
r
r
2
r
5
x +6x +14x +15x +9x+10
r
r
4
r
3
r
r
r
2
r
4
x +6x +11x +18x +9x+10
r
r
3
r
r
4
r r r
r
3
r
2
r
r
12x +7x +17x +9x+10
r
r
r
4
r r
r
3
r
2
11x +12x +13x +9x+10
r
r r r r
r
4
r
12x +12x +12x +9x+10
r
r
r
2
r
r r
r
r
3
r
2
r
10x +7x +19x +9x+10
r r r
r
r
r r r r
r
r
r
r
r
r
r
9x4 +15x3 +12x2 +9x+10
r
r
r
r
r
9x4 +12x3 +15x2 +9x+10
r r
r
r r
r
r
r
9x4 +12x3 +15x2 +9x+10
r
r
r
r r r
r r
r
4
r
r
r r r r
r
3
r
r
r
2
r
4
8x +15x +13x +9x+10
r
r
r
r
3
r
r
2
r
r
4
8x +12x +16x +9x+10
r
r
3
r
2
7x +15x +14x +9x+10
r r
r r r
r
r
r
r
4
r
3
r
r
2
r
r
4
6x +16x +14x +9x+10
r r r r r
r r r r
3
r
r
r
r
2
4
6x +15x +15x +9x+10
r
3
r
r
2
6x +7x +23x +9x+10
r r
r r r
r
r
r
4
r
3
r r r r r
r r r r r
r
r
2
r
r
4
5x +16x +15x +9x+10
r
3
r
r
2
r
4
5x +12x +19x +9x+10
r
3
r
r
2
4x +16x +16x +9x+10
r
r r r r r
r
r
r
r
r
r
4x4 +15x3 +17x2 +9x+10
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
12x3 +24x2 +9x+10
r
r
r
8
7
r
6
r
5
r
4
r
3
r
2
r
x +2x +3x +4x +5x +6x +7x +8x +9x +10x+11
r
r
r
9
r
8
r
7
r
6
5
r
4
r
r
3
r
r
2
2x +3x +4x +5x +6x +7x +8x +10x +10x+11
r
r
9
r
8
r
r
r
r
r
7
6
5
4
3
r
r
2
r
x +3x +4x +5x +6x +7x +9x +10x +10x+11
r
r
r
r
16x3 +20x2 +9x+10
r
r
7x3 +29x2 +9x+10
r
9
r
r
235 trees with 11 vertices:
10
r
r r r r r r
36x2 +9x+10
r
r
r r r r r r
r r r r r r r
r
r
3x4 +15x3 +18x2 +9x+10
r r r r r r
15x3 +21x2 +9x+10
r
r
x4 +12x3 +23x2 +9x+10
r r r r r r
r
r
r r r r r
2x4 +15x3 +19x2 +9x+10
r
r
3x4 +16x3 +17x2 +9x+10
r r r r r
r
r r r r
r r r r r
r
r
r
r
9
r
r
r
r
r
7
6
5
4
3
8
r
r
r
2
x +2x +4x +5x +6x +8x +9x +10x +10x+11
r
r
r
r
9
r
r
r
r
r
7
6
5
4
3
8
r
r
r
r
2
r
8
r
7
r
6
r
5
4
r
r
7
r
6
r
5
4
r
r
3
r
r
2
3x +4x +5x +6x +7x +8x +12x +10x+11
r
r
3
r
2
r
r
r
8
r
7
r
6
r
5
4
r
r
3
r
r
2
2x +5x +5x +6x +7x +9x +11x +10x+11
r
r
r
r
r
r
r
r
r
r
r
2x8 +4x7 +5x6 +6x5 +8x4 +10x3 +10x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
2x8 +3x7 +4x6 +8x5 +8x4 +9x3 +11x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
r
r
2x8 +3x7 +5x6 +6x5 +9x4 +9x3 +11x2 +10x+11
r
r
r
2x8 +4x7 +5x6 +6x5 +7x4 +10x3 +11x2 +10x+11
r
2x8 +3x7 +6x6 +6x5 +8x4 +9x3 +11x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
r
r
r
x8 +4x7 +6x6 +6x5 +7x4 +10x3 +11x2 +10x+11
r
r r
r
r
r
r
r
r
8
7
6
5
4
r
r
3
r
r
2
x +4x +5x +6x +7x +10x +12x +10x+11
r
r
r
r
r
r
8
7
6
5
4
r
r
r
r
r
r
8
7
6
5
4
r
r
3
r
r
2
x +3x +5x +7x +8x +10x +11x +10x+11
r
r
3
r
r
2
x +3x +5x +7x +9x +10x +10x +10x+11
r
r
r
r
r
r
r
8
7
6
5
4
r
r
3
r
r
2
x +3x +5x +6x +8x +11x +11x +10x+11
r
r
r
r
r
r
r
r
8
7
6
5
4
r
r
r
3
r
r
2
x +3x +4x +7x +9x +10x +11x +10x+11
r
r
8
r
7
r
6
r
5
r
r
r
r
8
7
6
5
4
r
3
r
r
2
x +2x +5x +6x +10x +10x +11x +10x+11
r
r
r
r
8
7
6
5
4
r
3
r
r
2
x +2x +5x +8x +9x +10x +10x +10x+11
r
r
r
r
r
8
7
6
5
4
r
r
3
r
r
2
x +2x +5x +6x +9x +10x +12x +10x+11
r r
r
r
r
r r
r
4
r
r
r
r
r
3
r
r
2
r
x +2x +4x +7x +9x +11x +11x +10x+11
r
4x +4x +5x +6x +7x +8x +11x +10x+11
r r
r
r
8
x +2x +3x +5x +7x +8x +9x +10x +10x+11
r
r
r
r
r
r
8
7
6
5
4
r
3
r
r
2
r
x +2x +3x +8x +9x +10x +12x +10x+11
r
r
r
r r
r
r
r
r
r
r
r
r
6x7 +5x6 +6x5 +7x4 +8x3 +13x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
r r r
r
r
r
r
r
r
r
r r
r
4x7 +4x6 +7x5 +9x4 +9x3 +12x2 +10x+11
r
r
4x7 +5x6 +6x5 +7x4 +8x3 +15x2 +10x+11
r
r
r
4x7 +6x6 +6x5 +8x4 +10x3 +11x2 +10x+11
r
4x7 +6x6 +6x5 +7x4 +10x3 +12x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
r
3x7 +7x6 +6x5 +7x4 +9x3 +13x2 +10x+11
r r
r
r
7
r
r
r
6
5
4
r
r
r
3
r
2
3x +5x +6x +9x +11x +11x +10x+11
r r
r
r
7
r
7
r
r
5
4
r
r
r
r
6
5
4
r r
r
r
3
r
r
2
3x +4x +8x +8x +9x +13x +10x+11
r
r
3
r
r
2
3x +5x +6x +7x +11x +13x +10x+11
r
6
r r
r
7
r
r
r
6
5
r
r
4
r
3
r
r
2
3x +4x +6x +10x +9x +13x +10x+11
r
r
r
7
r
r r
r
r
r
6
5
4
r
r
3
r
r
2
2x +7x +6x +7x +10x +13x +10x+11
r
7
r
r
r
r
6
5
4
r
r
3
r
r
2
2x +6x +7x +8x +11x +11x +10x+11
r
r
r
r
r
r
r
r
r
r
r
r
r
r
2x7 +6x6 +7x5 +7x4 +11x3 +12x2 +10x+11
r
r
r
r
r
r
r
r
2x7 +5x6 +8x5 +10x4 +10x3 +10x2 +10x+11
r
r
r
7
r
r
r
r
r
6
5
4
r
r
3
r
r
2
2x +5x +8x +9x +10x +11x +10x+11
r
7
r
r
r
r
6
5
4
r
r
r
3
r
r
2
2x +5x +7x +8x +11x +12x +10x+11
r
r
r
r
r
r
7
6
5
r
r r
r
r
4
r
3
r
r
2
2x +5x +6x +10x +10x +12x +10x+11
r
7
r
r
r
6
5
4
r
r
3
r
r
2
2x +5x +6x +8x +12x +12x +10x+11
r
r
7
r
r r
r
r
r
6
5
4
r
r
r
3
r
2
r
2x +5x +6x +7x +12x +13x +10x+11
r
r
r
r
7
6
5
r
4
r
r
3
r
2
r
2x +4x +8x +10x +10x +11x +10x+11
r
r
r
r
r
r
r
r
r
r
r
2x7 +4x6 +7x5 +10x4 +11x3 +11x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
2x7 +4x6 +7x5 +9x4 +11x3 +12x2 +10x+11
r
r
r
r
r
r
r
7
6
5
r
r
4
r
3
r
2
2x +4x +6x +10x +12x +11x +10x+11
r
r
r
r
6
5
4
r
r
r
r
r
6
5
4
r
3
r
r
2
r
r
r
7
6
5
r
r
r
6
5
r
r
3
r
r
r
r
r
4
r
r
r
3
r
2
2x +3x +6x +11x +10x +13x +10x+11
r
3
r r
4
r
r
2
r
2
2x +3x +7x +10x +11x +12x +10x+11
r r
7
r
2x +4x +6x +9x +12x +12x +10x+11
r
2x +3x +8x +9x +10x +13x +10x+11
r
r
7
r r
r
7
r
r
r
r
r
r
r
7
6
5
4
r
r
3
r
r
2
x +5x +8x +7x +11x +13x +10x+11
r r
r r r
r
r
r
r
r
7
6
5
4
r
r
3
r
r
2
x +5x +6x +7x +11x +15x +10x+11
r
7
r
6
r
5
r
7
r
6
r
5
r
4
r
3
r
r
2
x +4x +8x +10x +11x +11x +10x+11
r
r
r
r
r
r
7
6
5
4
r
3
r
r
2
r
r
r
r
4
x +4x +8x +10x +11x +11x +10x+11
r
r
r
r
7
r
6
r
r
5
r
r
2
x +4x +7x +9x +12x +12x +10x+11
r
4
r r
r
3
r
r
3
r
r
2
x +4x +7x +10x +12x +11x +10x+11
r
r
r
r
7
r
6
r
5
r
r
r
4
r
3
r
r
2
x +4x +6x +10x +11x +13x +10x+11
r
r
r
r r
r
r
r
r
r
r
r
x7 +4x6 +6x5 +8x4 +13x3 +13x2 +10x+11
r
r
r
r
r
r
r
r
r
r
x7 +3x6 +7x5 +11x4 +12x3 +11x2 +10x+11
r
r
r r
r
7
r
6
r
5
r
r
4
r
3
r
r
2
x +3x +7x +10x +12x +12x +10x+11
r
7
r
6
r
r
r
r
r
r
5
4
r
7
r
6
r
5
r
r
2
r
x +3x +6x +10x +13x +12x +10x+11
r
r
4
r
3
r
r
2
x +3x +6x +11x +11x +13x +10x+11
r
3
r r
r r
r
r
r
r
7
6
5
4
r
3
r
r
2
r
x +3x +6x +9x +13x +13x +10x+11
r
r
7
r
6
5
r r
r
r
r
4
r r r
r
3
r
r
r
2
x +2x +6x +10x +13x +13x +10x+11
r r
r
r
5
r
4
r
3
r
6
r
5
r
4
r
r
r
r
2
r
2
r r r
r
6
9x +6x +7x +8x +15x +10x+11
r
3
r
5
r
4
r
3
r
r
2
8x +6x +7x +8x +16x +10x+11
r
r r
r
r
r
r
r
r
r
r
r
6x6 +8x5 +8x4 +10x3 +13x2 +10x+11
r
r
r
r
r r
r
r
r
r
6x6 +8x5 +7x4 +10x3 +14x2 +10x+11
r r
r
r
6
r
r
r
5
4
r
r
3
r
r
r r
r
r
r
5
4
r
2
6
6x +7x +9x +11x +12x +10x+11
r
r
r
r
3
r
2
6x +7x +7x +11x +14x +10x+11
r r
r
r
r
r
r
6
r
r r
r
5
4
r
3
r
r
r
2
6x +5x +11x +9x +14x +10x+11
r
r
6
5
r
5
r
4
r
3
r
r
r
2
r
r
r
r
r
r
4x6 +8x5 +11x4 +10x3 +12x2 +10x+11
r
4
r
3
6
r
r
r
r
5
4
r
r
r
r
r
r
r
r
r
r
r
r
r
5
4
r
r
r
2
4x +8x +9x +12x +12x +10x+11
r
6
r
r
r
6
r
r
2
4x +8x +8x +12x +13x +10x+11
r
r
r
r
6
5
r
r
5
4
r
r
r r
r
6
r
r
5
4
r
3
r
r
2
r
r
r
3
r
r
2
4x +8x +7x +12x +14x +10x+11
r
r
r
4
r
r r
r r
r
r
4x +8x +8x +12x +13x +10x+11
r
r
3
r
r
r
3
r
4x6 +8x5 +10x4 +12x3 +11x2 +10x+11
r
r
r
2
r
r
r
2
4x +9x +7x +9x +16x +10x+11
r
r
r
5
r
r
r
3
r
r
6
5x +6x +7x +8x +19x +10x+11
r
r
4
r r r
r
6
r
5x +8x +11x +10x +11x +10x+11
r r r r
r
r
3
r
2
r
4x +6x +11x +12x +12x +10x+11
r
x +2x +6x +10x +11x +15x +10x+11
r r
r
6
r
7
r
r
6
5
r
r
r
r
4
r
3
r
2
r
4x +6x +11x +12x +12x +10x+11
r r r
r
r
r
r
r
r
r
r
r
r
r
r
4x6 +6x5 +10x4 +12x3 +13x2 +10x+11
r
r
r
r
r
r
r
r r r
r
r
r
4x6 +6x5 +7x4 +12x3 +16x2 +10x+11
r
r
r r
r
r
r
r
4x6 +4x5 +13x4 +10x3 +14x2 +10x+11
r
r
r
r
r
r
r r
r
r
r
r
4x6 +5x5 +11x4 +9x3 +16x2 +10x+11
r
r
r
4x6 +6x5 +10x4 +12x3 +13x2 +10x+11
r r r r
r
r
r
r r
r
r
r
r
r
r
3x6 +10x5 +7x4 +10x3 +15x2 +10x+11
r
r r
r
r
r
r
6
5
r
r r
r
4
r
3
r
r
2
3x +8x +11x +12x +11x +10x+11
r
r
6
5
r
r
r
4
r
r
3
r
2
3x +8x +11x +10x +13x +10x+11
r r
r
r
r
6
5
r
r
r
4
r
3
r
r
2
3x +8x +10x +12x +12x +10x+11
r
6
r r
r
r
r
r
5
4
r
r
r
3
r
2
3x +8x +8x +12x +14x +10x+11
r
r
r r
r
r
6
5
r
r
r
r
4
r r
r
3
r
r
2
3x +7x +10x +11x +14x +10x+11
r
r
6
5
r
r
5
r
r
r
2
r r
r
r r
r
r
3
3x +6x +12x +12x +12x +10x+11
r r
6
r
4
r
4
r
3
r
r
2
3x +6x +10x +14x +12x +10x+11
r
6
r
r
r
5
4
r
r
r
3
r
2
3x +6x +9x +14x +13x +10x+11
r r
r
r r r r
r
r
r
5
4
r r
r
r
r
r
6
5
6
r
r
3
r
r
2
3x +6x +7x +14x +15x +10x+11
r
6
r
5
r
4
r r
r
r
3
r
2
r
3x +5x +10x +13x +14x +10x+11
r
3
r
r
2
3x +5x +12x +13x +12x +10x+11
r
4
r
r
r
r
6
5
r r
r
r
4
r
3
r
2
r
3x +4x +13x +10x +15x +10x+11
r r
r
r
r
r
r
r
r
r
r
r
2x6 +9x5 +11x4 +11x3 +12x2 +10x+11
r
r
r r r
r
r
r
r
r
6
5
r
r
r
r r
r
r
4
r
3
r
r
2
2x +8x +11x +12x +12x +10x+11
r
6
r
r
r
r
5
4
r
r
6
5
r
r
3
r
r
2
r
r
r
4
r
r
r
r
6
5
r
r
6
5
r
r
r
r
r
r
4
r
r
r
3
r
r
2
2x +7x +11x +13x +12x +10x+11
r
6
r r
r
r
5
4
r
r
6
5
r
r
4
r
r
2
2x +7x +9x +13x +14x +10x+11
r
3
r
r
3
r r
r
6
r
r
r
r
5
4
r
r
3
r
r
r
2
2x +7x +9x +13x +14x +10x+11
r
r r
r
r
r r
r
r
r
2
2x +7x +10x +13x +13x +10x+11
r
r
r
r
2
r
r
r
r
3
2x +7x +12x +13x +11x +10x+11
r
4
r
2
r
r
r
3
2x +8x +10x +12x +13x +10x+11
r r
2x +8x +9x +12x +14x +10x+11
r
r
2x6 +9x5 +7x4 +11x3 +16x2 +10x+11
r
r
r
r
r
r
2x6 +6x5 +12x4 +12x3 +13x2 +10x+11
r
r
r
r
r
r
r
r
r
r
2x6 +6x5 +11x4 +14x3 +12x2 +10x+11
r
r
r
r
6
5
r
r
r
r
4
r
3
r r r
r
r
r
2
2x +6x +10x +14x +13x +10x+11
r
6
6
r
r
5
4
r
5
4
r
r
r
r
3
r
r
2
2x +6x +7x +14x +16x +10x+11
r
3
r
r
2
2x +6x +8x +14x +15x +10x+11
r r r r
r
r
r
r
6
5
r
r r
r
r
r
4
3
r
r
2
2x +5x +11x +13x +14x +10x+11
r
r
r
6
5
r r
r
r
r
4
r
3
r
r
2
2x +5x +10x +15x +13x +10x+11
r
r r
r
r
r
r
5
4
3
r
r
r r
r
r
r
6
5
r
6
r
r
r
2
2x +5x +9x +15x +14x +10x+11
r
r r
r
r
r
6
5
r
r
4
r
3
r
2
r
2x +4x +12x +14x +13x +10x+11
r
4
r
3
r
2
r
2x +4x +11x +14x +14x +10x+11
r r r
r
r
r r r
r
r
r
r
r
r
r
r
r
r
r
r
r
r
2x6 +3x5 +13x4 +11x3 +16x2 +10x+11
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r r
r r
r
r
r r
r
r
r
r
6
5
4
r
3
r
r
r
2
r
6
r
5
r
4
r
r
3
r
2
x +7x +12x +13x +12x +10x+11
x +6x +12x +14x +12x +10x+11
r
r
r
r
6
5
r
r
r
r
4
r
r
3
r
r
r
2
r
6
x +6x +12x +14x +12x +10x+11
5
r r
r
r
r
4
r
r
3
r
2
x +6x +11x +14x +13x +10x+11
r r
r
r
r
6
r
5
r
r r
r
4
r
3
r
r
r
2
x +6x +11x +14x +13x +10x+11
r
r
r
6
r
4
r
3
r
5
r
4
r
r
3
r
2
x +6x +11x +12x +15x +10x+11
r r r r
r r r
r
5
r
6
r r
r
r
r
2
r
6
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r
r
4
3
r
r
r
2
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r
r
r
r
r r
r
r
r
r
r
r
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r
r r
r
r
r
r
r
r
r
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r
r r r r
r
r
r
6
5
4
r r r
r
r
3
r
r
r
2
x +5x +8x +15x +16x +10x+11
r
r
6
5
r
r r
r
r
r
r
4
3
r
6
r
5
r
r
5
r
r
2
r
4
r
r
2
r r r r
r
r
r
6
5
4
r
r
3
r
r
2
x +4x +9x +16x +15x +10x+11
r r r r
r r r r
r
r
3
x +4x +11x +14x +15x +10x+11
x +4x +10x +16x +14x +10x+11
6
r
4
r
3
r
r
2
x +3x +10x +15x +16x +10x+11
r
r
6
5
r
r
4
r
3
r
r
2
x +2x +11x +12x +19x +10x+11
r r
r r
r
r
5
r r r
r
4
r
3
r
2
r
r
12x +7x +8x +18x +10x+11
r
r r r r
r
5
r
4
r
3
r
2
r r
r
10x +7x +8x +20x +10x+11
r
5
r
r
r
4
r
3
r
2
r
9x +11x +11x +14x +10x+11
r r
r r
r
r
r
r
r r
r
r
r
r
9x5 +9x4 +11x3 +16x2 +10x+11
r
r
r
r
r
r
r
r
r
r
r
8x5 +12x4 +12x3 +13x2 +10x+11
r
r
r
r r
r
r
r
8x5 +11x4 +12x3 +14x2 +10x+11
r
r r r
r
r
5
r
r
4
r
r
r
3
r
r
2
r
r
5
r r
r
r
r
r
4
r
5
8x +11x +10x +16x +10x+11
r
r
4
r
r
r
r
2
r
4
r
r
4
r
r
r
2
r
r
3
r
r
r
r
2
r
r
r
3
r r
r
r
r
5
4
r
2
r
r
r
r
r
r
r
2
r r
r
r
r
r
6x5 +7x4 +8x3 +24x2 +10x+11
r
r
r
r r r r
r
5
r r r r
r
4
r
r
3
r
r
2
5x +11x +13x +16x +10x+11
r
r
5
r
4
r
3
r
r
2
5x +11x +9x +20x +10x+11
r r
r r r r r
r
5
r
r
4
3
r
r
r
r
r
r
r
2
r
5
5x +7x +13x +20x +10x+11
r
r
4
r
3
r
2
4x +14x +14x +13x +10x+11
r r r
r
r
r
r
r
r r r
r
r
4x +13x +14x +14x +10x+11
5
5
4
r
3
r
2
r
r r
r
4
r
r
2
r
3
r r
r
r
r
4
r
r
3
r
2
r
r
2
r
r
5
4
r r r r
r
r
3
r
2
r
6x +9x +14x +16x +10x+11
5x5 +13x4 +13x3 +14x2 +10x+11
r
r
r
3
r
6x +10x +14x +15x +10x+11
r
r
6x +11x +14x +14x +10x+11
r r
r
r
r
4
5
r
3
r r r r r
r
2
6x +12x +14x +13x +10x+11
r
r
4
r
r
2
r
r
r
5
6x +11x +14x +14x +10x+11
6x +11x +12x +16x +10x+11
r
r
3
r
r r
r
5
r r
r
4
r
r
r
3
r
r r
r
4
8x +8x +12x +17x +10x+11
6x +12x +14x +13x +10x+11
6x +12x +12x +15x +10x+11
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r
5
r
r
r
3
r
r
r r
r
5
r r
r
r
2
r
r
r
r
3
8x +10x +10x +17x +10x+11
6x +13x +14x +12x +10x+11
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r
r r r r
r r
3
r r
r r r
r
4x +13x +10x +18x +10x+11
r
r
r r r
r
r
r
r
r r
r
r
r
r
r
r
r
r
r
4x5 +12x4 +16x3 +13x2 +10x+11
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r
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r
r
4
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r
r
r
4x5 +12x4 +16x3 +13x2 +10x+11
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r
r
r
r
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r
r
r
4x5 +11x4 +16x3 +14x2 +10x+11
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r r
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r
2
4x +11x +16x +14x +10x+11
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r
r
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r
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3
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r
2
4x +11x +16x +14x +10x+11
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r
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r
3
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2
4x +11x +14x +16x +10x+11
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r
r
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r
r
r
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r
r
r
r
2
4x +10x +16x +15x +10x+11
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r
r
r
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r
4
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3
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2
4x +10x +16x +15x +10x+11
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r r r r
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r
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3
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r
2
4x +10x +14x +17x +10x+11
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r
r
r
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4
r r r r r
r r r r
r
3
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r
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2
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r
3
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2
4x +7x +16x +18x +10x+11
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r
r
r r r
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r
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r
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r
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r
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3x +13x +15x +14x +10x+11
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r
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r
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r
3x5 +11x4 +17x3 +14x2 +10x+11
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3x +11x +15x +16x +10x+11
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r
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3x +10x +17x +15x +10x+11
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r
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r r r
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r
3x +9x +17x +16x +10x+11
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r r r r
r r r r r
r
3
r
r
2
3x +9x +17x +16x +10x+11
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r
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r
4
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3
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r
r
2
3x +7x +17x +18x +10x+11
r
5
r
4
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3
r
r
2
2x +11x +16x +16x +10x+11
r
r
r
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r r r r
r r r r
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r
3
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r
2
2x +11x +12x +20x +10x+11
r
5
r
4
r r r r
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3
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r
2
2x +10x +18x +15x +10x+11
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r
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r
3
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r
2
2x +10x +16x +17x +10x+11
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r r r r
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r
r r r r
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r
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r
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r r r r r
r
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r r r r
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r
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r
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r
r
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r
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r
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r r r
r r r r r
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r
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r
4
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3
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r
r
2
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r
r
4
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r
r r r
r
3
r
r
2
r
4
16x +14x +15x +10x+11
r
r
r
r
4
3
r r
r r
r
r
r
r
r
r r r r
r
4
r
r
3
r
r
r
r
2
14x +14x +17x +10x+11
13x +18x +14x +10x+11
r r
r r
r
r
r
4
r
3
r
2
r r r r
r
3
r
2
15x +8x +22x +10x+11
r
r
r
r
3
r r r r
4
r
16x +8x +21x +10x+11
r r r r
r
4
15x +14x +16x +10x+11
r
r
2
r
2
r
r
3
r
r r
r r r
r
r
2
r
r
r
4
12x +18x +15x +10x+11
r
r
3
r
2
12x +18x +15x +10x+11
r r r
r r
r
r r r r
r
r
r
r
r
12x4 +14x3 +19x2 +10x+11
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r
r r r r r
r
r
r
r r r
r
r
3
r
r
2
r
r
r
r
r
2
r
r
r
r
r
r
2
r
r
r
r
2
r
r
2
10x +14x +21x +10x+11
r r r r r r
r
r
r
4
r
3
r
r
2
9x +18x +18x +10x+11
r
4
r
3
r
r
2
9x +18x +18x +10x+11
r r r
r r r r
3
r
3
r
4
r
r r r r
9x +20x +16x +10x+11
r
r
r
4
10x +20x +15x +10x+11
r
3
r
11x4 +18x3 +16x2 +10x+11
r
3
r
r
r r r r r
r
4
r r r
r r
r
12x4 +8x3 +25x2 +10x+11
11x +14x +20x +10x+11
4
r
r r r
r
r
r
r r r r r
4
r
8x +20x +17x +10x+11
r
r
r
r
4
r r r
r r r r r
r
3
r
2
r
8x +18x +19x +10x+11
r
r
4
r
3
r
2
r
7x +20x +18x +10x+11
r r
r r r r r r
r
r
r r r r r r
r
r
r
r
7x4 +8x3 +30x2 +10x+11
r
r
r r r r
r
r
r
6x4 +20x3 +19x2 +10x+11
r
r
r
r
6x4 +20x3 +19x2 +10x+11
r r
r r r r
r r r r r r
r
r
r
4
r
3
r
r
2
r
r
4
r
3
r
3
r
r
2
r
r
r
r
r
r
4
4x +20x +21x +10x+11
r
3
r
3
r
2
5x +18x +22x +10x+11
r r r r r r
2
r
4
5x +20x +20x +10x+11
r r r r r r
r
r
4
6x +14x +25x +10x+11
r r r r r r
r r r r r r
r
r
2
r
r
4
4x +20x +21x +10x+11
r
3
r
2
3x +20x +22x +10x+11
r
r r r r r
r
r
4
r
3
r
r
r
2
r
r
r
r
2
r
r
2
8x +37x +10x+11
r
r
3
r
r
r
r
r
2
45x +10x+11
r
3
r
2
x +14x +30x +10x+11
r
2
r
3
r
2
14x +31x +10x+11
r r r r r r r r
r
r
4
r r r r r r r
18x +27x +10x+11
r r r r r r r
3
r
2
r r r r r r r
20x +25x +10x+11
r
r
3
2x +18x +25x +10x+11
r r r r r r r
3
r
4
3x +18x +24x +10x+11
r
r r r r r r
r r r r r r
r
r
r
References
[1] H. Hosoya, On some counting polynomials in chemistry, Discrete Appl. Math. 19 (1988), 239–
257.
[2] I. Gutman, Some properties of the Wiener polynomial , Graph Theory Notes New York 25
(1993), 13–18.
[3] A. A. Dobrynin, Construction of graphs with a palindromic Wiener polynomial , Vychisl. Sistemy
151 (1994), 37–54.
[4] A. A. Dobrynin, Graphs with palindromic Wiener polynomials, Graph Theory Notes New York
27 (1994), 50–54.
[5] M. Lepović, I. Gutman, A collective property of trees and chemical trees, J. Chem. Inf. Comput.
Sci. 38 (1998), 823–826.
[6] I. Gutman, E. Estrada, O. Ivanciuc, Some properties of the Wiener polynomial of trees, Graph
Theory Notes New York 36 (1999), in press.
[7] I. Gutman, Hosoya polynomial and the distance of the total graph of a tree, Publ. Elektrotehn.
Fak. (Beograd) Ser. Mat., in press.
[8] D. Stevanović, The Hosoya polynomial of composite graphs, Discrete Math., in press.
[9] G. Caporossi, A. A. Dobrynin, I. Gutman, P. Hansen, Trees with palindromic Hosoya polynomials, Graph Theory Notes New York, submitted.
[10] E. Estrada, O. Ivanciuc, I. Gutman, A. Gutierrez, L. Rodriguez, Extended Wiener indices. A
new set of descriptors for quantitative structure-property studies, New J. Chem. 22 (1998),
819–822.
HOSOJINI POLINOMI STABALA SA DO 11 ČVOROVA
Dragan Stevanovića i Ivan Gutmanb
a
b
Filozofski fakultet, Ćirila i Metodija 2, 18000 Niš, i
Prirodno–matematički fakultet, p.f. 60, 34000 Kragujevac
I Z V O D
Neka je G povezani graf. Ako je d(G, k) broj parova čvorova grafa G čije rastojanje je k , onda
P
se k≥0 d(G, k) xk naziva Hosojin polinom grafa G . Tabelirani su Hosojini polinomi svih stabala
sa 11 ili manje čvorova (1 + 1 + 1 + 2 + 3 + 6 + 11 + 23 + 47 + 106 + 235 stabala).