The Story of Zagreb Indices - UGentcaagt.ugent.be/csd5/slides/Nikolic_CSD5.pdf · n The Institute...

The Story of Zagreb IndicesSonja Nikolić

CSD 5 - Computers and Scientific Discovery 5

University of Sheffield, UK, July 20--23, 2010

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

mailto:[email protected]



Zagreb


Collaborators

n Nenad Trinajstićn The Rugjer Bošković Institute Zagreb,

Croatia

n Ante Miličevićn The Institute of Medical Research and

Occupational Health, Zagreb, Croatia


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


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


Introduction

n We applied a family of Zagreb indices to study molecules and complexity of selected classes of molecules


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


Graph

n Graphn verticesn edges

G

vertex

edge


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.


1

3

22

1

9

44

6

3

4

6

M1=18 M2=19


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.


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


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


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.


Modified reformulated Zagreb indices

mEM1 = Σ [d(ei) d(ei)]-1edges

mEM2 = Σ [d(ei) d(ej)]-1edges


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.


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


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


3

22

The cyclopropane subgraph

1716

1

3

22

Graph G as its own subgraph

1819


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


1

3

22G

1

11

111

11

∑ di*2(s) = 8

∑ di*·dj

* (s) = 4

TM1* = 100 TM2

* = 80


Modified Zagreb complexity indices

mTM1 = ∑ ∑ di-2 (s)

(s) vertices

mTM2 = ∑ ∑ (di·dj)-1 (s)

(s) edges


Variants of Modified Zagreb complexity indices

mTM1* = ∑ ∑ di

* -2 (s)(s) vertices

mTM2* = ∑ ∑ (di

* ·dj *) -1 (s)

(s) edges


G

mTM1 = 15.57mTM2 = 6.75mTM1

* = 29.72mTM2

* = 14.17


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)


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)


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


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


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.


Zagreb Matrices

M1 = ∑ [ ZM]iivertices

M2 = ∑ [ ZM]ijedges


Zagreb matrices

d(i) d(i) if i = j d(i) d(j) if vertices i and j are adjacent ij0 otherwise

=

ZM


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


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


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)


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.


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

http://www.pmf.kg.ac.yu/match/mcm3.htm


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.

http://www.iccmse.org/


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