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The Story of Zagreb IndicesSonja Nikolić
CSD 5 - Computers and Scientific Discovery 5
University of Sheffield, UK, July 20--23, 2010
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Sonja Sonja [email protected]@irb.hr
Rugjer Boskovic InstituteRugjer Boskovic InstituteBijenicka cesta 54, P.O.Box 180Bijenicka cesta 54, P.O.Box 180
10002 ZAGREB10002 ZAGREBCROATIACROATIA
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Collaborators
n Nenad Trinajstićn The Rugjer Bošković Institute Zagreb,
Croatia
n Ante Miličevićn The Institute of Medical Research and
Occupational Health, Zagreb, Croatia
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
n Measuring complexity in chemical systems, biological organisms or even poetry requires the counting of things.
n S.H. Bertz and W.F. Wrightn Graph Theory Notes of New York, 35 (1998)
32-48
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
The structure of the lecture
n Introduction n Original formulation of the Zagreb indicesn Modified Zagreb indicesn Variable Zagreb indicesn Reformulated original Zagreb indicesn Reformulated modified Zagreb indicesn Zagreb complexity indices n General Zagreb indicesn Zagreb indices for heterocyclic systemsn A variant of the Zagreb complexity indicesn Modified Zagreb complexity indices and their variantsn Zagreb coindices and outlinedn Properies of Zagreb indicesn Zagreb indices of line graphs n Zagreb co-indices n Analytical formulas for computing Zagreb indicesn Applicationn Conclusion
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Introduction
n We applied a family of Zagreb indices to study molecules and complexity of selected classes of molecules
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Motivation
n Zagreb indices, have been introduced 38 years ago (I. Gutman and N. Trinajstić, Chem. Phys. Lett. 17 (1972) 535-538) by Zagreb Group
n Current interest in Zagreb indices which found use in the QSPR/QSAR modeling (R. Todeschini and V. Consonni, Handbook of Molecular Descriptors, Wiley-VCH, Weinheim, 2009)
n Zagreb indices are included in a number of programs used for the routine computation of topological indices
n POLLYn DRAGONn CERIUSn TAMn DISSIM
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Graph
n Graphn verticesn edges
G
vertex
edge
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Original Zagreb indices
n M1 = ∑ di2 first Zagreb index
verticesn di = the degree of a vertex i
n M2 = ∑ di·dj second Zagreb indexedges
n di dj = the degree of a edge ij
I. Gutman and N. Trinajstić, Chem. Phys. Lett. 17 (1972) 535-538.I. Gutman, B. Ruščić, N. Trinajstić and C.F. Wilkox, Jr., J. Chem. Phys. 62
(1975) 3399-3405.
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
1
3
22
1
9
44
6
3
4
6
M1=18 M2=19
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb indices via squared adjacency vertex matrices
n M1 = ∑ (A2)ii (A2)iivertices
(A2)ii = d(i)n M2 = ∑ (A2)ii (A2)ii
edges
M. Barysz, D. Plavšić and N. Trinajstić, MATCHComm.Math. Chem. 19 (1986) 89-116.
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Modified Zagreb indicesS. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić,
Croat. Chem. Acta 76 (2003) 113.
n mM1 = ∑ di-1
vertices
n mM2 = ∑ (di·dj) -1edges
n mM2 = 1ON D. Bonchev, J. Mol. Graphics Modell.
20 (2001) 65.
1
0.11
0.250.25
0.17
0.33
0.25
0.17
mM1=1.61
mM2=0.92
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Variable Zagreb indicesA. Miličević, S. Nikolić, Croat. Chem. Acta 77 (2004) 97.
n λM1= ∑ di λ
vertices
n λM2 = ∑ (di·dj)λ
edges
λ= variable parameterλλ = 1 M1, M2
λλ = -1 mM1, mM2λ= -1/2 χ
λλMM11/V /V ≤≤ λλMM22/E/E
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Reformulated Zagreb indices
EM1 = Σ [d(ei) d(ei)]edges
EM2 = Σ [d(ei) d(ej)]edges
ei = degree of edge i
A. Miličević, S. Nikolić, N. Trinajstić, Mol. Diversity 8 (2004) 393.
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Modified reformulated Zagreb indices
mEM1 = Σ [d(ei) d(ei)]-1edges
mEM2 = Σ [d(ei) d(ej)]-1edges
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb complexity indices (2003)
n TM1 = ∑ ∑ di2 (s) = ∑ M1(s)
(s) vertices
n TM2 = ∑ ∑ di·dj (s) = ∑ M2(s)(s) edges
n Computation starts with the creation of the library containing all connected subgraphs of a molecular graph. Then each vertex in a subgraph is given the degree that the vertex possesses in the graph.
n Bonchev in 1997 originated this approach based on the subgraphs to construct topological indices
S. Nikolić, N. Trinajstić, I.M. Tolić, G. Rücker, C. Rücker, u: Complexity -Introduction and Fundamentals. D. Bonchev, D.H. Rouvray, editors, Taylor & Francis, London, 2003, str. 29-89.
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Example of the subgraph library
1
3
22G
1
3
22
1
33
222
23
TM1=230
TM2=145
The methane subgraphs
∑ di2(s)=18
i
∑di·dj (s)=0i
The ethane subgraphs
44
19
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
32
2
32
2
1 1
32
1
2
The butane subgraphs
36
26
The isobutane subgraph
1815
The propane subgraphs
1
32
1
32
2
3
2
32
2
32
2
79
50
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
3
22
The cyclopropane subgraph
1716
1
3
22
Graph G as its own subgraph
1819
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
A variant of the Zagreb complexityindices* (2003)
n TM1* = ∑ ∑ di
*2(s)(s) vertices
n di* = the degree of a vertex i as in a subgraph s
n s = the subgraph in G
n TM2* = ∑ ∑ di
* dj* (s)
(s) edges
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
1
3
22G
1
11
111
11
∑ di*2(s) = 8
∑ di*·dj
* (s) = 4
TM1* = 100 TM2
* = 80
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Modified Zagreb complexity indices
mTM1 = ∑ ∑ di-2 (s)
(s) vertices
mTM2 = ∑ ∑ (di·dj)-1 (s)
(s) edges
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Variants of Modified Zagreb complexity indices
mTM1* = ∑ ∑ di
* -2 (s)(s) vertices
mTM2* = ∑ ∑ (di
* ·dj *) -1 (s)
(s) edges
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
G
mTM1 = 15.57mTM2 = 6.75mTM1
* = 29.72mTM2
* = 14.17
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Application
n Note some criteria for complexity indicesn CI indices should increase (or decrease)
withn Molecular sizen Branchingn Cyclicityn And should be sensitive to symmetry
(optional)
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Chains
I K #A B C
D E F G H I
M 1 2 6 1 0 1 4 1 8 2 2 2 6 3 0 3 4M 2 1 4 8 1 2 1 6 2 0 2 4 2 8 3 2
m M 1 2 2 . 2 5 2 . 5 2 . 7 5 3 3 . 2 5 3 . 5 3 . 7 5 4m M 2 1 1 1 . 2 5 1 . 5 1 . 7 5 2 2 . 2 5 2 . 5 2 . 7 5T M 1 4 2 2 5 6 1 1 0 1 8 8 2 9 4 4 3 2 6 0 6 8 2 0
T M 1 * 2 1 0 2 8 6 0 1 1 0 1 8 2 2 8 0 4 0 8 5 7 0T M 2 1 8 2 8 6 4 1 2 0 2 0 0 3 0 8 4 4 8 6 2 4
T M 2 * 1 6 1 9 4 4 8 5 1 4 6 2 3 1 3 4 4 4 8 9m T M 1 4 7 1 1 1 6 . 2 5 2 3 3 1 . 5 0 4 2 5 4 . 7 5 7 0
m T M 1 * 2 6 . 2 5 1 3 2 2 . 5 0 3 5 5 0 . 7 5 7 0 9 3 1 2 0m T M 2 1 2 4 7 1 1 . 2 5 1 7 2 4 . 5 0 3 4 4 5 . 7 5
m T M 2 * 1 3 6 . 2 5 1 1 1 7 . 5 0 2 6 3 6 . 7 5 5 0 6 6t w c 2 1 0 3 2 8 8 2 2 2 5 3 6 1 2 5 4 2 8 7 8 6 5 0 0Ν T 3 6 1 0 1 5 2 1 2 8 3 6 4 5 5 5
Tests: total walk count twc (Rücker, Rücker, 2000)Total number of all connected subgraphs NT (Bonchev, 1997)
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Cycles
I K #
J K L M N O
M 1 = M 2 1 2 1 6 2 0 2 4 2 8 3 2m M 1 0 . 7 5 1 1 . 2 5 1 . 5 1 . 7 5 2m M 2 0 . 7 5 1 1 . 2 5 1 . 5 1 . 7 5 2T M 1 8 4 1 7 6 3 2 0 5 2 8 8 1 2 1 1 8
T M 1 * 3 6 8 8 1 8 0 3 2 4 5 3 2 8 1 6T M 2 4 8 1 1 2 2 2 0 3 8 4 6 1 6 9 2 8
T M 2 * 2 7 6 8 1 4 5 2 7 0 4 5 5 7 1 2m T M 1 5 . 2 5 1 1 2 0 3 3 5 0 . 7 5 7 4
m T M 1 * 1 3 . 5 2 8 4 8 . 7 5 7 6 . 5 1 1 2 1 5 6m T M 2 3 7 1 3 . 7 5 2 4 3 8 . 5 5 8
m T M 2 * 6 . 7 5 1 4 2 5 4 0 . 5 6 1 . 2 5 8 8N T 1 0 1 7 2 6 3 7 5 0 6 5
t w c 1 8 5 6 1 5 0 3 7 2 8 8 2 2 0 3 2
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Hexane trees
I K #
I I I I I I I V V
M 1 1 8 2 0 2 0 2 2 2 4M 2 1 6 1 8 1 9 2 1 2 2
m M 1 3 3 . 6 1 3 . 6 1 4 . 2 2 4 . 3 1m M 2 1 . 7 5 1 . 5 8 1 . 6 7 1 . 4 4 1 . 3 7T M 1 1 8 8 2 7 7 3 0 0 4 0 4 5 0 5
T M 1 * 1 1 0 1 4 6 1 5 8 1 9 6 2 2 2T M 2 1 2 0 1 7 2 1 9 9 2 6 4 2 9 0
T M 2 * 8 5 1 1 4 1 2 5 1 5 6 1 7 3m T M 1 2 3 3 3 . 5 3 3 5 4 8 . 4 4 5 5
m T M 1 * 3 5 4 4 . 3 3 4 7 . 4 4 5 7 . 3 9 6 4 . 1 5m T M 2 1 1 . 2 5 1 2 . 8 3 1 4 1 5 . 1 1 1 5 . 5 0
m T M 2 * 1 7 . 5 0 2 0 . 6 7 2 1 . 8 3 2 4 . 7 8 2 6 . 1 2t w c 2 2 2 2 6 8 2 8 4 3 3 0 3 7 0N T 2 1 2 4 2 5 2 8 3 0
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Overall Zagreb indicessOM1 = Σ Σ d(i)d(i) (s) = TM1
s i∈V
sOM2 = Σ Π d(i)d(j) (s) ≠ TM2s ij∈E
D. Bonchev, N. Trinajstic, SAR QSAR Environ. Res. 12 (2001) 213.
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb Matrices
M1 = ∑ [ ZM]iivertices
M2 = ∑ [ ZM]ijedges
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb matrices
d(i) d(i) if i = j d(i) d(j) if vertices i and j are adjacent ij0 otherwise
=
ZM
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Zagreb matrices of weighted graphs
d(i) d(i) if i = j2d(i) d(i) w if the vertex i is weighted
d(i) d(j) if vertices i and j are adjacent ijd(i) d(j) w if one vertex in the edge i-j is weigh
= ZMted
0 otherwise
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Example
1
6
2
3
4
5 1
1
3
2w
2
1
(a) (b)
1 3 0 0 0 0
3 9 6 0 0 320 6 4 0 04
0 0 4 4 2 0
0 0 0 2 1 0
0 3 0 0 0 1
=
ΖΜ
w
w www
w = weighted parameter
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Some properties of Zagreb indices
M1/V ≤ M2/E
Pierre Hansen valid for monocyclic graphs - Caporossi et al. (2010)
M1/V = M2/E = 4
all monocyclic graphs, Vukičević, Graovac, Hansen (2007, 2008)
vM1/V ≤ vM2/E
all graphs with v∈[0,1/2], Vukičević (2007)all chemical graphs with v∈[0,1]
all graphs v∈[-∞, 0], Huang et al. (2010) all monocyclic graphs v∈[1,+∞], Zhang, Liu (2010)
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Perspectives
Apparently, Zagreb indices as well as the family of all connectivity indices represent a mathematically-attractive invariants. Thus, we expect many more studies on these indices and look forward to further development of this area of matematical chemistry.
X. Li and I. Gutman, Mathematical Aspects of Randić-type Molecular Structure Descriptors, University of Kragujevac, Kragujevac, Serbia, 2006.
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
MATHEMATICAL CHEMISTRY MONOGRAPHS, No. 3
Publisher: University of Kragujevac and Faculty of Science Kragujevac
http://www.pmf.kg.ac.yu/match/mcm3.htmD. Janezic, A. Milicevic, S. Nikolic, and
N. TrinajsticGraph-Theoretical Matrices in Chemistry
2007, VI + 205 pp., Hardcover, ISBN: 86-81829-72-6
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University of Sheffield, UK, July 19-23, 2010 S. NIKOLIĆ: A Story of Zagreb Indices
Eighth International Conference of Computational Methods in Sciences and Engineering - ICCMSE 2010Psalidi, Kos, Greece, 03-08 October 2010http://www.iccmse.org/
Symposium 4Title: 8th Symposium on Mathematical ChemistryOrganizer: Dr. Sonja Nikolic, The Rugjer Boskovic
Institute, Zagreb, CroatiaEnquiries and contributions to E-mail: [email protected]
Scope and Topics: Graph theory development, studying complexity of molecules and reactions, development of molecular descriptors, development of mathematical invariants of chemical and biological systems, modelling structure-property-activity, advanced chemometrics and chemoinformaticsalgorithms as the tools required by chemical engineers and analytical chemists to explore their data and build predictive models.