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علماء الرياضيات | Hosoya Index |
 1436
 03:10 مساءً
date: 7-4-2022 |
Author : Devillers, J. and Balaban, A. T. 
Book or Source : Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach 
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Date: 27-3-2022
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Date: 3-4-2022
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Date: 14-4-2022
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As proposed by Hosoya (1971), the Hosoya index (also called 
-index) of a graph is defined by
 |
 |
 |
(1) |
 |
 |
 |
(2) |
where 
is the number of vertices of the graph, 
is the 
th coefficient of the matching polynomial, 
is the 
th coefficient of the matching-generating polynomial, and 
is the absolute value of 
. In others words, it is just the number of independent edge sets (i.e., matchings) in a graph.
An alternate definition for the Hosoya index defined by Devillers and Balaban (1999, p. 105) is given by
 |
(3) |
where 
denotes the floor function. This definition is identical to 
except for graphs with odd vertex count, in which case it is 0 (making it not terribly useful).
Unless otherwise stated, hydrogen atoms are usually ignored in the computation of such indices as organic chemists usually do when they write a benzene ring as a hexagon (Devillers and Balaban 1999, p. 25).
The following table summarizes values of the Hosoya index for various special classes of graphs.
| graph class | OEIS |  ,  , ... |
| Andrásfai graph | A000000 | 2, 11, 106, 1475, 27514, 651815, 18926340, 655968971, ... |
| antiprism graph | A192742 | X, X, 51, 191, 708, 2631, 9775, 36319, 134943, 501380, ... |
| Apollonian network | A000000 | 10, 99, 38613, ... |
cocktail party graph  |
A000000 | 1, 7, 51, 513, 6345, 93255, 1584555, 30524865, 656843985, ... |
complete bipartite graph  |
A002720 | 2, 7, 34, 209, 1546, ... |
complete graph  |
A000085 | 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35696, 140152, ... |
complete tripartite graph  |
A000000 | 4, 51, 1126, 37201, 1670136, 96502339, ... |
| crossed prism graph | A000000 | X, 108, 1092, 11208, 115272, ... |
| crown graph | A144085 | X, X, 18, 108, 780, 6600, 63840, 693840, 8361360, ... |
cycle graph  |
A000032 | X, X, 4, 7, 11, 18, 29, 47, 76, 123, 199, 322, 521, ... |
empty graph  |
A000012 | 1, 1, 1, 1, 1, 1, 1, 1, ... |
| folded cube graph | A000000 | 2, 10, 209, 115536, 85609174977, ... |
grid graph  |
A028420 | 1, 7, 131, 10012, 2810694, 2989126727, 11945257052321, ... |
grid graph  |
A033535 | 1, 1, 108, 49793133, 17312701462385916505, ... |
| halved cube graph | A000000 | 1, 2, 10, 513, 4281761, ... |
hypercube graph  |
A045310 | 2, 7, 108, 41025, 13803794944, ... |
Keller graph  |
A000000 | 1, 115536, ... |
Möbius ladder  |
A020877 | X, X, 34, 106, 344, 1102, 3546, ... |
| Mycielski graph | A000000 | 1, 2, 11, 968, 37270256, ... |
odd graph  |
A000000 | 1, 4, 332, 11311777344, ... |
| pan graph | A006355 | 6, 10, 16, 26, 42, 68, 110, 178, 288, 466, ... |
path graph  |
A000045 | 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, ... |
permutation star graph  |
A000000 | 1, 2, 18, 1157484, ... |
prism graph  |
A102080 | X, X, 32, 108, 342, 1104, 3544, 11396, 36626, ... |
rook graph  |
A000000 | 1, 7, 370, 270529, 3337807996, ... |
star graph  |
A000027 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, ... |
| sun graph | A192856 | X, X, 27, 100, 393, 1624, 7017, 31558, 147177, ... |
sunlet graph  |
A002203 | X, X, 14, 34, 82, 198, 478, 1154, 2786, 6726, ... |
torus grid graph  |
A000000 | X, X, 370, 40125, ... |
transposition graph  |
A000000 | 1, 2, 34, 161966673, ... |
| triangular graph | A000000 | 1, 4, 51, 2460, 513619, 509709696, ... |
| web graph | A192857 | X, X, 93, 439, 1988, 9107, 41583, 190047, 868341, 3967828, ... |
wheel graph  |
A061705 | X, X, X, 10, 19, 36, 66, 120, 215, 382, 673, 1178, 2050, 3550, 6121, ... |
Closed forms are summarized in the following table, where 
denotes the 
th polynomial root of 
, 
is a confluent hypergeometric function of the second kind, 
is a Lucas number, 
is a Laguerre polynomial, 
is a Fibonacci number, and 
is a Pell-Lucas number.
| graph |  |
| antiprism graph |  |
complete graph  |
 |
complete bipartite graph  |
 |
cycle graph  |
 |
empty graph  |
1 |
Möbius ladder  |
 |
| pan graph |  |
path graph  |
 |
prism graph  |
 |
star graph  |
 |
sunlet graph  |
 |
wheel graph  |
 |
Devillers, J. and Balaban, A. T. (Eds.). Topological Indices and Related Descriptors in QSAR and QSPR. Amsterdam, Netherlands: Gordon and Breach, pp. 27-28 and 105, 1999.
Hosoya, H. "A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons." Bull. Chem. Soc. Japan 44, 2332-2339, 1971.
Hosoya, H. and Murakami, M. "Topological Index as Applied to 
-Electronic Systems. II. Topological Bond Order." Bull. Chem. Soc. Japan 48, 3512-3517, 1975.
Sloane, N. J. A. Sequences A000012/M0003, A000027/M0472, A000085/M1221, A000045/M0692, A002203, A002720/M1795, A006355, A020877, A025169, A028420, A033535, A045310, A102080, A144085, A192742, A192856, A192857, and A192858 in "The On-Line Encyclopedia of Integer Sequences."
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