Polynomial Functions Topic 2: Equations of Polynomial Functions.
Hosoya polynomial of small trees
23
ZBORNIK RADOVA Prirodno–matematiˇ ckog 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 Stevanovi´ c a and Ivan Gutman b a Faculty of Philosophy, ´ Cirila i Metodija 2, 18000 Niˇ s, 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 distance k apart, then ∑ k≥0 d(G, k) x k is called the Hosoya polynomial of G . The Hosoya polynomials of 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 k=0 d(G, k) x k
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Hosoya polynomial of small trees
Transcript of Hosoya polynomial of small trees
ZBORNIK RADOVA Prirodno–matematickog 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 Stevanovica and Ivan Gutmanb
aFaculty of Philosophy, Cirila i Metodija 2, 18000 Nis, Yugoslavia, andbFaculty 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
distance k apart, then∑
k≥0d(G,k) xk is called the Hosoya polynomial of G . The Hosoya polynomials of
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∑
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,
namely as∑
k≥1d(G, k)xk . Even in this case the constant term d(G, 0) is fully determined: because
of the identityD∑
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: 1 tree with 3 vertices:
r
1
r r
x+2
r rr
x2+2x+3
2 trees with 4 vertices:
r rr r
x3+2x2+3x+4
r
r rr
3x2+3x+4
3 trees with 5 vertices:
r rr rr
x4+2x3+3x2+4x+5
r r
r
r r
2x3+4x2+4x+5
r r
r rr
6x2+4x+5
6 trees with 6 vertices:
r rr rrr
x5+2x4+3x3+4x2+5x+6
r
r rr rr
2x4+3x3+5x2+5x+6
r
r rr rr
x4+4x3+5x2+5x+6
rr
r rrr
4x3+6x2+5x+6
r r
r r
r r
3x3+7x2+5x+6
r r r
r rr
10x2+5x+6
11 trees with 7 vertices:
r rr rrr r
x6+2x5+3x4+4x3+5x2+6x+7
r
r rr rrr
2x5+3x4+4x3+6x2+6x+7
r
r rr r rr
x5+3x4+5x3+6x2+6x+7
rr
r rr rr
4x4+4x3+7x2+6x+7
r
r r
r
r rr
3x4+6x3+6x2+6x+7
r r
r rr rr
3x4+4x3+8x2+6x+7
r
r
r
r
r r r
2x4+6x3+7x2+6x+7
r r
r rr rr
x4+6x3+8x2+6x+7
r
r r
r r
r r
6x3+9x2+6x+7
r r
r r r
r r
4x3+11x2+6x+7
r r r r
r rr
15x2+6x+7
23 trees with 8 vertices:
r rr rrr r r
x7+2x6+3x5+4x4+5x3+6x2+7x+8
r
r rr rrrr
2x6+3x5+4x4+5x3+7x2+7x+8
r
r rr r rrr
x6+3x5+4x4+6x3+7x2+7x+8
r
r rrrr rr
x6+2x5+5x4+6x3+7x2+7x+8
rr
r rr r rr
4x5+4x4+5x3+8x2+7x+8
r r
r rr rrr
3x5+4x4+5x3+9x2+7x+8
r
r r
r
r r rr
2x5+5x4+7x3+7x2+7x+8
r
r r
rr rr r
2x5+5x4+6x3+8x2+7x+8
r
r r
rr rr r
2x5+4x4+7x3+8x2+7x+8
r r
r rrr r r
x5+4x4+8x3+8x2+7x+8
r r
r rr r rr
x5+4x4+7x3+9x2+7x+8
r
r r
r r
rr r
6x4+5x3+10x2+7x+8
r r
r rr r
r
r
5x4+8x3+8x2+7x+8
r rr
r rrr r
4x4+8x3+9x2+7x+8
r r r
r rr rr
4x4+5x3+12x2+7x+8
r
r
r r
r
r rr
3x4+9x3+9x2+7x+8
r
r
r
r r
r r r
3x4+8x3+10x2+7x+8
r r
rr r
r rr
2x4+9x3+10x2+7x+8
r r r
r rr rr
x4+8x3+12x2+7x+8
rr r
r r
r
rr
9x3+12x2+7x+8
r
r r
r r r
r r
8x3+13x2+7x+8
r r
r r r r
r r
5x3+16x2+7x+8
r r r r r
r rr
21x2+7x+8
47 trees with 9 vertices:
r rr rrr r rr
x8+2x7+3x6+4x5+5x4+6x3+7x2+8x+9
r
r rr rrr rr
2x7+3x6+4x5+5x4+6x3+8x2+8x+9
r
r rr r rrrr
x7+3x6+4x5+5x4+7x3+8x2+8x+9
r
r rrr r rr r
x7+2x6+4x5+6x4+7x3+8x2+8x+9
rr
r rr rr rr
4x6+4x5+5x4+6x3+9x2+8x+9
r r
r rr rrrr
3x6+4x5+5x4+6x3+10x2+8x+9
r
r r
rrr rr r
2x6+5x5+5x4+7x3+9x2+8x+9
r
r r
r
r r rrr
2x6+4x5+6x4+8x3+8x2+8x+9
r
r r
rr rrr r
2x6+4x5+5x4+8x3+9x2+8x+9
r
r r
rr r rr r
2x6+3x5+7x4+7x3+9x2+8x+9
r
r r
r
rrr rr
x6+4x5+7x4+8x3+8x2+8x+9
r r
r rrr rr r
x6+4x5+6x4+8x3+9x2+8x+9
r r
r rr r rrr
x6+4x5+5x4+8x3+10x2+8x+9
r r
r rr r rr r
x6+3x5+6x4+9x3+9x2+8x+9
r r
r rrrr rr
x6+2x5+7x4+8x3+10x2+8x+9
r
r r
r r
r rr r
6x5+5x4+6x3+11x2+8x+9
r
r r
r
r r
r
r r
4x5+7x4+8x3+9x2+8x+9
r
r r
rr
rr rr
4x5+6x4+8x3+10x2+8x+9
r r r
r rr rrr
4x5+5x4+6x3+13x2+8x+9
r r
r rr r r
r
r
3x5+7x4+9x3+9x2+8x+9
rr r
r rr rrr
3x5+7x4+7x3+11x2+8x+9
rr r
r rr rrr
3x5+5x4+9x3+11x2+8x+9
r
r r
r
r r
r
r r
2x5+7x4+10x3+9x2+8x+9
r
r r r
rr rr r
2x5+7x4+8x3+11x2+8x+9
r
r r
r
r
r r r r
2x5+6x4+10x3+10x2+8x+9
r
r
r r
r
r r rr
2x5+6x4+10x3+10x2+8x+9
r
r r r
rr rr r
2x5+5x4+10x3+11x2+8x+9
rr r
r rrr rr
x5+5x4+11x3+11x2+8x+9
r r r
r rr r rr
x5+5x4+9x3+13x2+8x+9
rr r
r r
r
r rr
9x4+6x3+13x2+8x+9
r
rr
r r
r
r rr
8x4+10x3+10x2+8x+9
r
r r
r r r
rr r
8x4+6x3+14x2+8x+9
r r r
r rr rr
r
7x4+10x3+11x2+8x+9
r r
r r
r r
r rr
6x4+12x3+10x2+8x+9
r r
r r
rr
r rr
6x4+10x3+12x2+8x+9
r r
r
r rr r
r
r
5x4+12x3+11x2+8x+9
r r r r
r rr rr
5x4+6x3+17x2+8x+9
r r rr
r rrr r
4x4+12x3+12x2+8x+9
r
r
r
r r r
r r r
4x4+10x3+14x2+8x+9
r r
r
r r
r
r rr
3x4+12x3+13x2+8x+9
r
r r
r
r r
r r r
3x4+12x3+13x2+8x+9
r r
rr r r
r rr
2x4+12x3+14x2+8x+9
r r r r
r rr rr
x4+10x3+17x2+8x+9
r r
r r
r r r
r r
12x3+16x2+8x+9
r
r r
r r r r
r r
10x3+18x2+8x+9
r r
r r r r r
r r
6x3+22x2+8x+9
r r r r r r
r rr
28x2+8x+9
106 trees with 10 vertices:
r rr rrr r rrr
x9+2x8+3x7+4x6+5x5+6x4+7x3+8x2+9x+10
r
r rr rrr r rr
2x8+3x7+4x6+5x5+6x4+7x3+9x2+9x+10
r
r rr r rr rrr
x8+3x7+4x6+5x5+6x4+8x3+9x2+9x+10
r
r rrr r rr rr
x8+2x7+4x6+5x5+7x4+8x3+9x2+9x+10
r
r rr rr rrr r
x8+2x7+3x6+6x5+7x4+8x3+9x2+9x+10
rr
r rr rrr rr
4x7+4x6+5x5+6x4+7x3+10x2+9x+10
r r
r rr rrr rr
3x7+4x6+5x5+6x4+7x3+11x2+9x+10
r
r r
rrr rrr r
2x7+5x6+5x5+6x4+8x3+10x2+9x+10
r
r r
r
r r rrrr
2x7+4x6+5x5+7x4+9x3+9x2+9x+10
r
r r
rr rrrr r
2x7+4x6+5x5+6x4+9x3+10x2+9x+10
r
r r
rr rr rr r
2x7+3x6+6x5+7x4+8x3+10x2+9x+10
r
r r
rr r rrr r
2x7+3x6+5x5+8x4+8x3+10x2+9x+10
r r
r rrr rrr r
x7+4x6+6x5+6x4+9x3+10x2+9x+10
r r
r rr r rrrr
x7+4x6+5x5+6x4+9x3+11x2+9x+10
r
r r
r
rr r rr r
x7+3x6+6x5+8x4+9x3+9x2+9x+10
r r
r rrr r rr r
x7+3x6+5x5+8x4+9x3+10x2+9x+10
r r
r rr r rrr r
x7+3x6+5x5+7x4+10x3+10x2+9x+10
rr
r rrr r rrr
x7+2x6+5x5+8x4+10x3+10x2+9x+10
r r
r rrr r rr r
x7+2x6+5x5+8x4+9x3+11x2+9x+10
r
r r
r r
r rrr r
6x6+5x5+6x4+7x3+12x2+9x+10
r
r r
r
r
r
r rr r
4x6+6x5+7x4+9x3+10x2+9x+10
r
r r
r
r
r rr r r
4x6+6x5+6x4+9x3+11x2+9x+10
r r r
r rr rrrr
4x6+5x5+6x4+7x3+14x2+9x+10
r rr
r rr r rr r
4x6+4x5+9x4+8x3+11x2+9x+10
rr r
r rrrr rr
3x6+7x5+6x4+8x3+12x2+9x+10
r
r r
r
rr
r
r rr
3x6+6x5+9x4+9x3+9x2+9x+10
r r
r rr r rr
r
r
3x6+5x5+8x4+10x3+10x2+9x+10
rr r
r rr rrrr
3x6+5x5+6x4+10x3+12x2+9x+10
rr r
r rr r rrr
3x6+4x5+9x4+8x3+12x2+9x+10
r
r r r
rrr rr r
2x6+7x5+6x4+9x3+12x2+9x+10
r
r
r
rr
r
r rr r
2x6+6x5+9x4+9x3+10x2+9x+10
r
r r
r
rr r
r
r r
2x6+6x5+8x4+10x3+10x2+9x+10
r
r r r
rr rr r r
2x6+6x5+7x4+10x3+11x2+9x+10
r
r rr
rrrr r r
2x6+5x5+8x4+10x3+11x2+9x+10
r
r
r r
r
r r rrr
2x6+5x5+7x4+11x3+11x2+9x+10
r
r r r
rr rrr r
2x6+5x5+6x4+11x3+12x2+9x+10
r
r
r r
r
r r rrr
2x6+4x5+9x4+11x3+10x2+9x+10
r
r
r r
rr rrr r
2x6+4x5+8x4+11x3+11x2+9x+10
r
r r r
rr r rr r
2x6+3x5+10x4+9x3+12x2+9x+10
r r
r rrrr r
r
r
x6+6x5+9x4+10x3+10x2+9x+10
r
r
r rr
r
r rr r
x6+5x5+9x4+11x3+10x2+9x+10
r r r
r rrr rr r
x6+5x5+8x4+10x3+12x2+9x+10
r r r
r rr r rrr
x6+5x5+6x4+10x3+14x2+9x+10
r
r
r r
r
rrr rr
x6+4x5+9x4+11x3+11x2+9x+10
rr r
r rrr rr r
x6+4x5+8x4+12x3+11x2+9x+10
rr r
r rr r rrr
x6+4x5+7x4+12x3+12x2+9x+10
r r r
r rr r rr r
x6+3x5+8x4+12x3+12x2+9x+10
r r r
r rrrr rr
x6+2x5+9x4+10x3+14x2+9x+10
rr r
r r
r
r r rr
9x5+6x4+7x3+14x2+9x+10
r
r r
r r r
r rr r
8x5+6x4+7x3+15x2+9x+10
r r
r
r rr r
r
r r
6x5+9x4+10x3+11x2+9x+10
r r
r r
r
r r
r
rr
6x5+9x4+9x3+12x2+9x+10
rr
r r
r r
rrr r
6x5+8x4+9x3+13x2+9x+10
r r
r r
rr
rr rr
6x5+7x4+10x3+13x2+9x+10
r r r r
r rr rrr
5x5+6x4+7x3+18x2+9x+10
r
r r
r
r
r
r
r r r
4x5+10x4+12x3+10x2+9x+10
r
r r
r rr rr
r
r
4x5+10x4+11x3+11x2+9x+10
r
r r
r
r
r
r
rr r
4x5+9x4+12x3+11x2+9x+10
r
r r
r
r
r r
r
r r
4x5+9x4+11x3+12x2+9x+10
r r r
r rr r rr
r
4x5+9x4+11x3+12x2+9x+10
rr r r
r rr rrr
4x5+9x4+8x3+15x2+9x+10
rr
r r
rr
rr r r
4x5+8x4+12x3+12x2+9x+10
r
r r
rr r
rr rr
4x5+8x4+11x3+13x2+9x+10
rr r r
r rr rrr
4x5+6x4+11x3+15x2+9x+10
r r
r
r rrr
r
rr
3x5+10x4+12x3+11x2+9x+10
r
r r r r
rr rr r
3x5+10x4+9x3+14x2+9x+10
r r
r r
r r
r r rr
3x5+9x4+13x3+11x2+9x+10
r r
r
r rr r r
r
r
3x5+8x4+13x3+12x2+9x+10
r
r r
r
r r
r r r r
3x5+8x4+12x3+13x2+9x+10
r
r r r r
rr rr r
3x5+6x4+13x3+14x2+9x+10
r r
r r
r
r
r
r rr
2x5+9x4+13x3+12x2+9x+10
r
r r r r
rr rr r
2x5+9x4+10x3+15x2+9x+10
r r
r r
r
r r
r
r r
2x5+8x4+14x3+12x2+9x+10
r
r r
r
rr
r r rr
2x5+8x4+13x3+13x2+9x+10
r
r
r
rr r
r r rr
2x5+7x4+14x3+13x2+9x+10
r r
r
r r
r
r r rr
2x5+7x4+13x3+14x2+9x+10
r
r r r r
rr rr r
2x5+6x4+13x3+15x2+9x+10
r r rr
r rrr r r
x5+6x4+15x3+14x2+9x+10
rr r r
r rrr rr
x5+6x4+14x3+15x2+9x+10
r r r r
r rr r rr
x5+6x4+11x3+18x2+9x+10
r r
rr
r r
r
rr r
12x4+12x3+12x2+9x+10
r r
r r
r r r
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r
r
r r
r rr
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r
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r rr
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r
r
9x4+15x3+12x2+9x+10
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r rr r
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r rr
9x4+12x3+15x2+9x+10
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rr r
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r
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r
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r r r r r
r rr rr
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r r
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r rr r
r
r
5x4+16x3+15x2+9x+10
r
r
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r
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r
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r r
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r rr
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235 trees with 11 vertices:
r rr rrr r rrr r
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r
r rr rrr r rrr
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r rrr r rr rrr
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r rrr r rr r rr
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rr
r rr rrr r rr
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r r
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4x7+6x6+6x5+7x4+10x3+12x2+10x+11
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4x7+5x6+6x5+7x4+8x3+15x2+10x+11
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4x7+4x6+7x5+9x4+9x3+12x2+10x+11
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3x7+7x6+6x5+7x4+9x3+13x2+10x+11
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rr rr
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rr r r r
r rr rrr
5x5+11x4+9x3+20x2+10x+11
rr r r r
r rr rrr
5x5+7x4+13x3+20x2+10x+11
r
r
r
r r
r
r r r r
r
4x5+14x4+14x3+13x2+10x+11
r r r
r
r rrr r r
r
4x5+13x4+14x3+14x2+10x+11
r rr r r
r rr rrr
4x5+13x4+10x3+18x2+10x+11
r rr
r r
r
r r r
r
r
4x5+12x4+16x3+13x2+10x+11
r
r
r r
r
r
r r
r
r r
4x5+12x4+16x3+13x2+10x+11
r
r
r r
r
r
r
r
rr r
4x5+11x4+16x3+14x2+10x+11
r
r r r
r
r
r
r
r rr
4x5+11x4+16x3+14x2+10x+11
rr
r r
r rr rr
r
r
4x5+11x4+16x3+14x2+10x+11
r
r r r
r
r
r r
r
r r
4x5+11x4+14x3+16x2+10x+11
r rr
r r
rr
r r rr
4x5+10x4+16x3+15x2+10x+11
r r r
r
r rr r r
r
r
4x5+10x4+16x3+15x2+10x+11
r
r r
rr r r
rr rr
4x5+10x4+14x3+17x2+10x+11
r
r r
r
r r r
r r r r
4x5+10x4+14x3+17x2+10x+11
r rr r r
r rr rrr
4x5+7x4+16x3+18x2+10x+11
r r
r r
r rrr
r
rr
3x5+13x4+15x3+14x2+10x+11
r
r r r r r
rr rr r
3x5+13x4+11x3+18x2+10x+11
r
r r
r r
r r
r r
r r
3x5+12x4+17x3+13x2+10x+11
r r
rr
r rrr
r
rr
3x5+11x4+17x3+14x2+10x+11
r
rr r
r
r r
r rr r
3x5+11x4+15x3+16x2+10x+11
r
r r
r r
r r
r r rr
3x5+10x4+17x3+15x2+10x+11
r
r r r
r
r r
r r r r
3x5+9x4+17x3+16x2+10x+11
r r
r r
r rr r r
r
r
3x5+9x4+17x3+16x2+10x+11
r
r r r r r
rr rr r
3x5+7x4+17x3+18x2+10x+11
r r r
r r
r
r
r
r rr
2x5+11x4+16x3+16x2+10x+11
r
r r r r r
rr rr r
2x5+11x4+12x3+20x2+10x+11
r r r
r r
r
r r
r
r r
2x5+10x4+18x3+15x2+10x+11
r
r r r
r
rr
r r rr
2x5+10x4+16x3+17x2+10x+11
r r r
r r
r
r r
r
r r
2x5+9x4+18x3+16x2+10x+11
r r
rr r r r
r rr r
2x5+9x4+18x3+16x2+10x+11
r
r
r
rr r r
r r rr
2x5+8x4+18x3+17x2+10x+11
r r r
r
r r
r
r r rr
2x5+8x4+16x3+19x2+10x+11
r
r r r r r
rr rr r
2x5+7x4+16x3+20x2+10x+11
r rr r r
r rrr rr
x5+7x4+19x3+18x2+10x+11
rr r r r
r rrr rr
x5+7x4+17x3+20x2+10x+11
r r r r r
r rr r rr
x5+7x4+13x3+24x2+10x+11
r r r
rr
r rrr r
r
16x4+14x3+15x2+10x+11
rr r r
r r
r r
r rr
16x4+8x3+21x2+10x+11
r
rr r
r r
rr
r rr
15x4+14x3+16x2+10x+11
r r
r r
r r r r
rr r
15x4+8x3+22x2+10x+11
r
r
r r
r r rr
rr r
14x4+14x3+17x2+10x+11
r r rr
r rr r
r r
r
13x4+18x3+14x2+10x+11
r r rr
r r
r
r r
r
r
12x4+18x3+15x2+10x+11
r
r r
rr
r r
r
rr r
12x4+18x3+15x2+10x+11
rr r
r r
r r r
rr r
12x4+14x3+19x2+10x+11
r
r r
r r r r r
rr r
12x4+8x3+25x2+10x+11
r r
rr
r r r
r rr
r
11x4+18x3+16x2+10x+11
r r r r r
r rr rr
r
11x4+14x3+20x2+10x+11
r r r
r r
r r r
r rr
10x4+20x3+15x2+10x+11
r r r r
r r
rr
r rr
10x4+14x3+21x2+10x+11
r r
r
r
r r
r
r r
r
r
9x4+20x3+16x2+10x+11
r
r r r r
r rr rr
r
9x4+18x3+18x2+10x+11
r r rr r
r r
r
rr r
9x4+18x3+18x2+10x+11
r
r r rr
r r
r
rr r
8x4+20x3+17x2+10x+11
r
r r r
r
r r r
r r r
8x4+18x3+19x2+10x+11
r r r
r r
r rr r
r
r
7x4+20x3+18x2+10x+11
r r r r r r
r rr rr
7x4+8x3+30x2+10x+11
r
r r
r rr r r
r rr
6x4+20x3+19x2+10x+11
r r
r r
r r
r r
r rr
6x4+20x3+19x2+10x+11
r
r
r
r r r r r
r r r
6x4+14x3+25x2+10x+11
r r
r r r
r rr r
r
r
5x4+20x3+20x2+10x+11
r
r r
r
r r r r
r r r
5x4+18x3+22x2+10x+11
r r r r rr
r rrr r
4x4+20x3+21x2+10x+11
r
r r r
r
r r r
r r r
4x4+20x3+21x2+10x+11
r r
r rr r r r
r rr
3x4+20x3+22x2+10x+11
r r r r
r
r r
r
r rr
3x4+18x3+24x2+10x+11
r r
rr r r r r
r rr
2x4+18x3+25x2+10x+11
r r r r r r
r rr rr
x4+14x3+30x2+10x+11
r r r
r r
r r r r
r r
20x3+25x2+10x+11
r r
r r
r r r r r
r r
18x3+27x2+10x+11
r
r r
r r r r r r
r r
14x3+31x2+10x+11
r r
r r r r r r r
r r
8x3+37x2+10x+11
r r r r r r r r
r rr
45x2+10x+11
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. Lepovic, 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. Stevanovic, The Hosoya polynomial of composite graphs , Discrete Math., in press.
[9] G. Caporossi, A. A. Dobrynin, I. Gutman, P. Hansen, Trees with palindromic Hosoya polyno-
mials , 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 CVOROVA
Dragan Stevanovica i Ivan Gutmanb
aFilozofski fakultet, Cirila i Metodija 2, 18000 Nis, i
bPrirodno–matematicki fakultet, p.f. 60, 34000 Kragujevac
I Z V O D
Neka je G povezani graf. Ako je d(G, k) broj parova cvorova grafa G cije rastojanje je k , onda
se∑
k≥0d(G, k)xk naziva Hosojin polinom grafa G . Tabelirani su Hosojini polinomi svih stabala
sa 11 ili manje cvorova (1 + 1 + 1 + 2 + 3 + 6 + 11 + 23 + 47 + 106 + 235 stabala).
![Notes on Polynomial Functors - UAB Barcelonakock/cat/polynomial.pdf · 2018. 1. 11. · • Polynomial functors and polynomial monads [39] with Gambino • Polynomial functors and](https://static.fdocuments.us/doc/165x107/60faf8a63b5d714a860ca184/notes-on-polynomial-functors-uab-barcelona-kockcat-2018-1-11-a-polynomial.jpg)
