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

The Story of Zagreb I ndices Sonja Nikoli ć 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 Institute Bijenicka cesta 54, P.O.Box 180 Bijenicka cesta 54, P.O.Box 180 10002 ZAGREB 10002 ZAGREB CROATIA CROATIA S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Zagreb S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Zagreb S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Collaborators n Nenad Trinajsti ć n The Rugjer Boškovi ć Institute Zagreb, Croatia n Ante Mili č evi ć n The Institute of Medical Research and Occupational Health, Zagreb, Croatia S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

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 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

The structure of the lecture 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 n S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Introduction n We applied a family of Zagreb indices to study molecules and complexity of selected classes of molecules S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Motivation Zagreb indices, have been introduced 38 years ago (I. n Gutman and N. Trinajsti ć , Chem. Phys. Lett. 17 (1972) 535-538) by Zagreb Group Current interest in Zagreb indices which found use in the n QSPR/QSAR modeling (R. Todeschini and V. Consonni , Handbook of Molecular Descriptors , Wiley-VCH, Weinheim, 2009) Zagreb indices are included in a number of programs n used for the routine computation of topological indices n POLLY n DRAGON n CERIUS n TAM n DISSIM S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Graph n Graph n vertices n edges edge vertex G S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Original Zagreb indices n M 1 = ∑ d i 2 first Zagreb index vertices n d i = the degree of a vertex i n M 2 = ∑ d i ·d j second Zagreb index edges n d i d j = the degree of a edge ij I . Gutman and N. Trinajsti ć , Chem. Phys. Lett. 17 (1972) 535-538. I. Gutman, B. Ruš č i ć , N. Trinajsti ć and C.F. Wilkox, Jr., J. Chem. Phys. 62 (1975) 3399-3405. S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

1 3 2 2 1 3 9 6 6 4 4 4 M 1 =18 M 2 =19 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Zagreb indices via squared adjacency vertex matrices n M 1 = ∑ ( A 2 ) ii ( A 2 ) ii vertices ( A 2 ) ii = d(i) n M 2 = ∑ ( A 2 ) ii ( A 2 ) ii edges M. Barysz, D. Plavši ć and N. Trinajsti ć , MATCH Comm.Math. Chem. 19 (1986) 89-116. S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Modified Zagreb indices S. Nikoli ć , G. Kova č evi ć , A. Mili č evi ć , N. Trinajsti ć , Croat. Chem. Acta 76 (2003) 113. 1 m M 1 = ∑ d i -1 n 0.11 vertices 0.25 0.25 m M 2 = ∑ (d i ·d j ) -1 m M 1 =1.61 n edges 0.33 0.17 m M 2 = 1 ON 0.17 n D. Bonchev , J. Mol. Graphics Modell. 0.25 m M 2 =0.92 20 (2001) 65. S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Variable Zagreb indices A. Mili č evi ć , S. Nikoli ć , Croat. Chem. Acta 77 (2004) 97. n λ M 1 = ∑ d i λ λ = variable parameter λ = 1 λ M 1 , M 2 vertices λ = -1 λ m M 1 , m M 2 λ = -1/2 χ n λ M 2 = ∑ (d i ·d j ) λ ≤ λ λ λ M /V ≤ λ M M 1 M 2 /E 1 /V 2 /E edges S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Reformulated Zagreb indices EM 1 = Σ [ d (e i ) d (e i ) ] edges EM 2 = Σ [ d (e i ) d (e j ) ] edges e i = degree of edge i A. Mili č evi ć , S. Nikoli ć , N. Trinajsti ć , Mol. Diversity 8 (2004) 393. S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Modified reformulated Zagreb indices m EM 1 = Σ [ d (e i ) d (e i ) ] -1 edges m EM 2 = Σ [ d (e i ) d (e j ) ] -1 edges S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Zagreb complexity indices (2003) n TM 1 = ∑ ∑ d i 2 (s) = ∑ M 1 (s) (s) vertices n TM 2 = ∑ ∑ d i ·d j (s) = ∑ M 2 (s) (s) edges Computation starts with the creation of the library containing all connected n subgraphs of a molecular graph. Then each vertex in a subgraph is given the degree that the vertex possesses in the graph. Bonchev in 1997 originated this approach based on the subgraphs to n construct topological indices S. Nikoli ć , N. Trinajsti ć , I.M. Toli ć , G. Rücker, C. Rücker , u: Complexity - Introduction and Fundamentals. D. Bonchev, D.H. Rouvray, editors, Taylor & Francis, London, 2003, str. 29-89. S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Example of the subgraph library 1 TM 1 = 230 3 TM 2 = 145 2 2 G ∑ d i 1 2 (s)= 18 The methane i subgraphs 3 ∑ d i ·d j (s)= 0 2 2 i 1 44 The ethane 3 3 3 2 2 subgraphs 19 2 2 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

79 1 1 3 3 The propane 2 3 3 2 3 2 2 50 subgraphs 2 2 2 2 1 1 36 The butane 3 3 2 2 subgraphs 26 2 2 The isobutane 1 18 subgraph 3 15 2 2 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

3 The cyclopropane 17 subgraph 16 2 2 1 Graph G as its 18 3 own subgraph 19 2 2 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

A variant of the Zagreb complexity indices* (2003) * = ∑ ∑ d i * 2 (s) n TM 1 (s) vertices * = the degree of a vertex i as in a subgraph s n d i n s = the subgraph in G * = ∑ ∑ d i * d j * (s) n TM 2 (s) edges S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

1 1 3 1 1 1 1 1 2 2 1 1 G ∑ d i * 2 (s) = 8 ∑ d i * (s) = 4 * ·d j * = 100 * = 80 TM 1 TM 2 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Modified Zagreb complexity indices m TM 1 = ∑ ∑ d i -2 (s) (s) vertices -1 (s) m TM 2 = ∑ ∑ (d i ·d j ) (s) edges S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Variants of Modified Zagreb complexity indices * -2 (s) * = ∑ ∑ d i m TM 1 (s) vertices -1 (s) * ·d j * = ∑ ∑ (d i * ) m TM 2 (s) edges S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

m TM 1 = 15.57 m TM 2 = 6.75 * = 29.72 m TM 1 G * = 14.17 m TM 2 S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

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) S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010

Chains I K # A B C D E F G H I M 2 6 1 0 1 4 1 8 2 2 2 6 3 0 3 4 1 M 1 4 8 1 2 1 6 2 0 2 4 2 8 3 2 2 m M 2 2 . 2 5 2 . 5 2 . 7 5 3 3 . 2 5 3 . 5 3 . 7 5 4 1 m M 1 1 1 . 2 5 1 . 5 1 . 7 5 2 2 . 2 5 2 . 5 2 . 7 5 2 T M 4 2 2 5 6 1 1 0 1 8 8 2 9 4 4 3 2 6 0 6 8 2 0 1 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 T M 1 8 2 8 6 4 1 2 0 2 0 0 3 0 8 4 4 8 6 2 4 2 2 * 1 6 1 9 4 4 8 5 1 4 6 2 3 1 3 4 4 4 8 9 T M m T M 4 7 1 1 1 6 . 2 5 2 3 3 1 . 5 0 4 2 5 4 . 7 5 7 0 1 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 1 2 4 7 1 1 . 2 5 1 7 2 4 . 5 0 3 4 4 5 . 7 5 2 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 3 6 1 0 1 5 2 1 2 8 3 6 4 5 5 5 T Tests: total walk count twc (Rücker, Rücker, 2000) Total number of all connected subgraphs N T (Bonchev, 1997) S. NIKOLI Ć : A Story of Zagreb Indices University of Sheffield, UK, July 19-23, 2010