The Story of Zagreb I ndices Sonja Nikoli CSD 5 - Computers and - - PowerPoint PPT Presentation

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The Story of Zagreb I ndices Sonja Nikoli CSD 5 - Computers and Scientific Discovery 5 University of Sheffield, UK , July 20--23, 2010 Sonja Nikolic Nikolic Sonja sonja@irb.hr sonja@irb.hr Rugjer Boskovic Institute Rugjer Boskovic

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The Story of Zagreb I ndices

Sonja 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 Nikolic Nikolic sonja@irb.hr sonja@irb.hr Rugjer Boskovic Institute Rugjer Boskovic Institute Bijenicka cesta 54, P.O.Box 180 Bijenicka cesta 54, P.O.Box 180 10002 ZAGREB 10002 ZAGREB CROATIA CROATIA

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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 Trinajstić

n The Rugjer Bošković Institute Zagreb,

Croatia

n Ante Miličević

n 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. Wright

n 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 indices

n

Modified Zagreb indices

n

Variable Zagreb indices

n

Reformulated original Zagreb indices

n

Reformulated modified Zagreb indices

n

Zagreb complexity indices

n

General Zagreb indices

n

Zagreb indices for heterocyclic systems

n

A variant of the Zagreb complexity indices

n

Modified Zagreb complexity indices and their variants

n

Zagreb coindices and outlined

n

Properies of Zagreb indices

n

Zagreb indices of line graphs

n

Zagreb co-indices

n

Analytical formulas for computing Zagreb indices

n

Application

n

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 POLLY n DRAGON n CERIUS n TAM n DISSIM

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University of Sheffield, UK, July 19-23, 2010

  • S. NIKOLIĆ: A Story of Zagreb Indices

Graph

n Graph

n vertices n 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 = ∑ di

2

first Zagreb index

vertices

n di = the degree of a vertex i

n M2 = ∑ di·dj

second Zagreb index

edges

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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  • S. NIKOLIĆ: A Story of Zagreb Indices

1 3 2 2 1 9 4 4 6 3 4 6 M1=18 M2=19

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  • S. NIKOLIĆ: A Story of Zagreb Indices

Zagreb indices via squared adjacency vertex matrices

n M1 = ∑ (A2)ii (A2)ii

vertices

(A2)ii = d(i)

n M2 = ∑ (A2)ii (A2)ii

edges

  • M. Barysz, D. Plavšić and N. Trinajstić, MATCH

Comm.Math. Chem. 19 (1986) 89-116.

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  • S. NIKOLIĆ: A Story of Zagreb Indices

Modified Zagreb indices

  • S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić,
  • Croat. Chem. Acta 76 (2003) 113.

n

mM1 = ∑ di

  • 1

vertices n

mM2 = ∑ (di·dj) -1

edges n

mM2 = 1ON

  • D. Bonchev, J. Mol. Graphics Modell.

20 (2001) 65.

1 0.11 0.25 0.25 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 indices

  • A. Miličević, S. Nikolić, Croat. Chem. Acta 77 (2004) 97.

n λM1= ∑ di

λ

vertices n λM2 = ∑ (di·dj)λ edges

λ= variable parameter λ λ = 1 M1, M2 λ λ = -1

mM1, mM2

λ= -1/2 χ

λ λM

M1

1/V

/V ≤

≤ λ

λM

M2

2/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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  • S. NIKOLIĆ: A Story of Zagreb Indices

Modified reformulated Zagreb indices

mEM1 = Σ [d(ei) d(ei)]-1

edges

mEM2 = Σ [d(ei) d(ej)]-1

edges

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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 = ∑

∑ di

2 (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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  • S. NIKOLIĆ: A Story of Zagreb Indices

Example of the subgraph library

1 3 2 2 G

1 3 2 2

1 3 3 2 2 2 2 3

TM1= 230 TM2= 145 The methane subgraphs

∑ di

2(s)= 18

i

∑di·dj (s)= 0

i The ethane subgraphs 44 19

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  • S. NIKOLIĆ: A Story of Zagreb Indices

3 2 2 3 2 2 1 1 3 2 1 2

The butane subgraphs 36 26 The isobutane subgraph 18 15 The propane subgraphs

1 3 2 1 3 2 2 3 2 3 2 2 3 2 2

79 50

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  • S. NIKOLIĆ: A Story of Zagreb Indices

3 2 2

The cyclopropane subgraph 17 16

1 3 2 2

Graph G as its

  • wn subgraph

18 19

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  • S. NIKOLIĆ: A Story of Zagreb Indices

A variant of the Zagreb complexity indices* (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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  • S. NIKOLIĆ: A Story of Zagreb Indices

1 3 2 2 G 1 1 1 1 1 1 1 1

∑ 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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  • S. NIKOLIĆ: A Story of Zagreb Indices

G

mTM1 = 15.57 mTM2 = 6.75 mTM1 * = 29.72 mTM2 * = 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 indices n CI indices should increase (or decrease)

with

n Molecular size n Branching n Cyclicity n And should be sensitive to symmetry

(optional)

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  • 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 4 M

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 4

m M 2

1 1 1 . 2 5 1 . 5 1 . 7 5 2 2 . 2 5 2 . 5 2 . 7 5 T 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 0 T 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 9

m 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 0

m 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 6 t 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 0 Ν

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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  • 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 2

m M 1

0 . 7 5 1 1 . 2 5 1 . 5 1 . 7 5 2

m M 2

0 . 7 5 1 1 . 2 5 1 . 5 1 . 7 5 2 T 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 6 T 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 2

m 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 6

m 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 8 N

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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  • 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 4 M

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 1

m M 2

1 . 7 5 1 . 5 8 1 . 6 7 1 . 4 4 1 . 3 7 T 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 2 T 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 3

m 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 5

m 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 2 t w c 2 2 2 2 6 8 2 8 4 3 3 0 3 7 0 N

T

2 1 2 4 2 5 2 8 3 0

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  • S. NIKOLIĆ: A Story of Zagreb Indices

Overall Zagreb indices

sOM1 = Σ

Σ d(i)d(i) (s) = TM1

s i∈V

sOM2 = Σ Π d(i)d(j) (s) ≠ TM2

s ij∈E

  • D. Bonchev, N. Trinajstic, SAR QSAR Environ.
  • Res. 12 (2001) 213.
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  • S. NIKOLIĆ: A Story of Zagreb Indices

Zagreb Matrices

M1 = ∑ [ ZM] ii

vertices

M2 = ∑ [ ZM] ij

edges

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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 ij 0 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 = j 2 d(i) d(i) w if the vertex i is weighted d(i) d(j) if vertices i and j are adjacent ij d(i) d(j) w if one vertex in the edge i-j is weigh =

   

ZM ted 0 otherwise

        

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  • S. NIKOLIĆ: A Story of Zagreb Indices

Example

1 6 2 3 4 5 1 1 3 2w 2 1 (a) (b)

1 3 3 9 6 3 2 6 4 4 4 4 2 2 1 3 1 =

                 

ΖΜ w w w w w

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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  • 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

  • f 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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  • S. NIKOLIĆ: A Story of Zagreb Indices

MATHEMATI CAL CHEMI STRY MONOGRAPHS, No. 3

Publisher: University of Kragujevac and Faculty of Science Kragujevac http://www.pmf.kg.ac.yu/match/mcm3.htm

  • D. Janezic, A. Milicevic, S. Nikolic, and
  • N. Trinajstic

Graph-Theoretical Matrices in Chemistry 2007, VI + 205 pp., Hardcover, ISBN: 86-81829-72-6

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  • S. NIKOLIĆ: A Story of Zagreb Indices

Eighth I nternational Conference of Computational Methods in Sciences and Engineering - I CCMSE 2010 Psalidi, Kos, Greece, 03-08 October 2010

http://www.iccmse.org/

Symposium 4 Title: 8th Symposium on Mathematical Chemistry Organizer: Dr. Sonja Nikolic, The Rugjer Boskovic I nstitute, Zagreb, Croatia Enquiries and contributions to E-mail: sonja@irb.hr Scope and Topics: Graph theory development, studying complexity of molecules and reactions, development

  • f molecular descriptors, development of

mathematical invariants of chemical and biological systems, modelling structure-property-activity, advanced chemometrics and chemoinformatics algorithms as the tools required by chemical engineers and analytical chemists to explore their data and build predictive models.