The Story of Zagreb Indices Sonja Nikolić CSD 5 - Computers and Scientific Discovery 5 University of Sheffield, UK, July 0--3, 00
Sonja Nikolic sonja@irb.hr Rugjer Boskovic Institute Bijenicka cesta 54, P.O.Box 80 000 ZAGREB CROATIA
Zagreb
Zagreb
Nenad Trinajstić Collaborators The Rugjer Bošković Institute Zagreb, Croatia Ante Miličević The Institute of Medical Research and Occupational Health, Zagreb, Croatia
Measuring complexity in chemical systems, biological organisms or even poetry requires the counting of things. S.H. Bertz and W.F. Wright Graph Theory Notes of New York, 35 (998) 3-48
The structure of the lecture Introduction Original formulation of the Zagreb indices Modified Zagreb indices Variable Zagreb indices Reformulated original Zagreb indices Reformulated modified Zagreb indices Zagreb complexity indices General Zagreb indices Zagreb indices for heterocyclic systems A variant of the Zagreb complexity indices Modified Zagreb complexity indices and their variants Zagreb coindices and outlined Properies of Zagreb indices Zagreb indices of line graphs Zagreb co-indices Analytical formulas for computing Zagreb indices Application Conclusion
Introduction We applied a family of Zagreb indices to study molecules and complexity of selected classes of molecules
Motivation Zagreb indices, have been introduced 38 years ago (I. Gutman and N. Trinajstić, Chem. Phys. Lett. 7 (97) 535-538) by Zagreb Group 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, 009) Zagreb indices are included in a number of programs used for the routine computation of topological indices POLLY DRAGON CERIUS TAM DISSIM
Graph Graph vertices edges edge vertex G
Original Zagreb indices M = d i first Zagreb index vertices d i = the degree of a vertex i M = d i d j second Zagreb index edges d i d j = the degree of a edge ij I. Gutman and N. Trinajstić, Chem. Phys. Lett. 7 (97) 535-538. I. Gutman, B. Ruščić, N. Trinajstić and C.F. Wilkox, Jr., J. Chem. Phys. 6 (975) 3399-3405.
3 4 3 9 6 6 4 4 M =8 M =9
Zagreb indices via squared adjacency vertex matrices M = (A ) ii (A ) ii vertices (A ) ii = d(i) M = (A ) ii (A ) ii edges M. Barysz, D. Plavšić and N. Trinajstić, MATCH Comm.Math. Chem. 9 (986) 89-6.
Modified Zagreb indices S. Nikolić, G. Kovačević, A. Miličević, N. Trinajstić, Croat. Chem. Acta 76 (003) 3. m M = d i - vertices m M = (d i d j ) - edges 0. 0.5 0.5 m M =.6 0.33 m M = ON D. Bonchev, J. Mol. Graphics Modell. 0 (00) 65. 0.7 0.7 0.5 m M =0.9
Variable Zagreb indices A. Miličević, S. Nikolić, Croat. Chem. Acta 77 (004) 97. λ M = d i λ vertices λ M = (d i d j ) λ λ= variable parameter λ = M, M λ = - m M, m M λ= -/ χ λ M /V λ M /E edges
Reformulated Zagreb indices EM = Σ[d(e i ) d(e i )] edges EM = Σ[d(e i ) d(ej)] edges e i = degree of edge i A. Miličević, S. Nikolić, N. Trinajstić, Mol. Diversity 8 (004) 393.
Modified reformulated Zagreb indices m EM = Σ [d(e i ) d(e i )] - edges m EM = Σ [d(e i ) d(e j )] - edges
Zagreb complexity indices (003) TM = d i (s) = M (s) (s) vertices TM = d i d j (s) = M (s) (s) edges 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. Bonchev in 997 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, 003, str. 9-89.
Example of the subgraph library TM =30 G 3 TM =45 The methane subgraphs 3 d i (s)=8 i d i d j (s)=0 i The ethane subgraphs 3 3 3 44 9
The propane subgraphs 3 3 3 3 3 79 50 The butane subgraphs 3 3 36 6 The isobutane subgraph 3 8 5
The cyclopropane subgraph 3 7 6 Graph G as its own subgraph 3 8 9
A variant of the Zagreb complexity indices* (003) TM * = d i * (s) (s) vertices d i* = the degree of a vertex i as in a subgraph s s = the subgraph in G TM * = (s) d i* d j* (s) edges
G 3 d i * (s) = 8 d i* d j* (s) = 4 TM * = 00 TM * = 80
Modified Zagreb complexity indices m TM = (s) m TM = d i - (s) vertices (d i d j ) - (s) (s) edges
Variants of Modified Zagreb complexity indices m TM * = * - d i (s) (s) vertices m TM * = (s) edges (d i * d j * ) - (s)
m TM = 5.57 m TM = 6.75 G m TM * = 9.7 m TM * = 4.7
Application Note some criteria for complexity indices CI indices should increase (or decrease) with Molecular size Branching Cyclicity And should be sensitive to symmetry (optional)
Chains IK # A B C D E F G H I M 6 0 4 8 6 30 34 M 4 8 6 0 4 8 3 m M.5.5.75 3 3.5 3.5 3.75 4 m M.5.5.75.5.5.75 TM 4 56 0 88 94 43 606 80 TM * 0 8 60 0 8 80 408 570 TM 8 8 64 0 00 308 448 64 TM * 6 9 44 85 46 3 344 489 m TM 4 7 6.5 3 3.50 4 54.75 70 m TM * 6.5 3.50 35 50.75 70 93 0 m TM 4 7.5 7 4.50 34 45.75 m TM * 3 6.5 7.50 6 36.75 50 66 tw c 0 3 88 536 54 878 6500 Ν T 3 6 0 5 8 36 45 55 Tests: total walk count twc (Rücker, Rücker, 000) Total number of all connected subgraphs N T (Bonchev, 997)
Cycles IK # J K L M N O M =M 6 0 4 8 3 m M 0.75.5.5.75 m M 0.75.5.5.75 TM 84 76 30 58 8 8 TM * 36 88 80 34 53 86 TM 48 0 384 66 98 TM * 7 68 45 70 455 7 TM 5.5 0 33 50.75 74 m TM * 3.5 8 48.75 76.5 56 TM 3 7 3.75 4 38.5 58 m TM * 6.75 4 5 40.5 6.5 88 N T 0 7 6 37 50 65 tw c 8 56 50 37 88 03
Hexane trees IK # I II III IV V M 8 0 0 4 M 6 8 9 M 3 3.6 3.6 4. 4.3 M.75.58.67.44.37 TM 88 77 300 404 505 TM * 0 46 58 96 TM 0 7 99 64 90 TM * 85 4 5 56 73 TM 3 33.53 35 48.44 55 m TM * 35 44.33 47.44 57.39 64.5 TM.5.83 4 5. 5.50 m TM * 7.50 0.67.83 4.78 6. tw c 68 84 330 370 N T 4 5 8 30
Overall Zagreb indices s OM = Σ Σ d(i)d(i) (s) = TM s i V s OM = Σ Π d(i)d(j) (s) TM s ij E D. Bonchev, N. Trinajstic, SAR QSAR Environ. Res. (00) 3.
Zagreb Matrices M = [ ZM] ii vertices M = [ ZM] ij edges
Zagreb matrices d(i) d(i) if i = j ZM = d(i) d(j) if vertices i and j are adjacent ij 0 otherwise
Zagreb matrices of weighted graphs ZM ij = d(i) d(i) d(i) d(i) w d(i) d(j) d(i) d(j) w 0 otherwise if i = j if the vertex i is weighted if vertices i and j are adjacent if one vertex in the edge i-j is weighted
Example 6 3 (a) 4 5 3 w (b) ΖΜ = 3 0 0 0 0 3 9 6w 0 0 3 0 6w 4w 4w 0 0 0 0 4w 4 0 0 0 0 0 0 3 0 0 0 w = weighted parameter
Some properties of Zagreb indices M /V M /E Pierre Hansen valid for monocyclic graphs - Caporossi et al. (00) M /V = M /E = 4 all monocyclic graphs, Vukičević, Graovac, Hansen (007, 008) v M /V v M /E all graphs with v [0,/], Vukičević (007) all chemical graphs with v [0,] all graphs v [-, 0], Huang et al. (00) all monocyclic graphs v [,+ ], Zhang, Liu (00)
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 Randićtype Molecular Structure Descriptors, University of Kragujevac, Kragujevac, Serbia, 006.
MATHEMATICAL CHEMISTRY 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 007, VI + 05 pp., Hardcover, ISBN: 86-889-7-6
Eighth International Conference of Computational Methods in Sciences and Engineering - ICCMSE 00 Psalidi, Kos, Greece, 03-08 October 00 http://www.iccmse.org/ Symposium 4 Title: 8 th Symposium on Mathematical Chemistry Organizer: Dr. Sonja Nikolic, The Rugjer Boskovic Institute, Zagreb, Croatia Enquiries and contributions to E-mail: sonja@irb.hr 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 chemoinformatics algorithms as the tools required by chemical engineers and analytical chemists to explore their data and build predictive models.