On atom–bond connectivity molecule structure descriptors J. Serb. Chem. Soc. 81 (3) 271–276 (2016) UDC 547.593:539.216:541.57–123:544.131 JSCS–4845 Original Scientific paper 271 On atom–bond connectivity molecule structure descriptors BORIS FURTULA1*, IVAN GUTMAN1,2# and KINKAR CH. DAS3 1Faculty of Science, University of Kragujevac, P. O. Box 60, 34000 Kragujevac, Serbia 2State University of Novi Pazar, Novi Pazar, Serbia and 3Department of Mathematics, Sungkyunkwan University, Suwon 440–746, South Korea (Received 1 September, accepted 12 October 2015) Abstract: The atom–bond connectivity index (ABC) is a degree-based mole- cular structure descriptor with well-documented chemical applications. In 2010, a distance-based new variant of this index (ABCGG) was proposed. Hitherto, the relation between ABC and ABCGG has not been analyzed. In this paper, the basic characteristics of this relation are established. In particular, ABC and ABCGG are not correlated and both cases > GGABC ABC and < GGABC ABC may occur in the case of (structurally similar) molecules. However, in the case of benzenoid hydrocarbons, ABC always exceeds ABCGG. Keywords: atom–bond connectivity index; ABC index; molecular structure descriptor; molecular graph. INTRODUCTION One of the most prolific areas of application of graph theory in chemistry is via molecular structure descriptors (topological indices), namely quantities that are calculated from the molecular graphs and that are used for modeling physico- chemical, pharmacological, toxicological, and other properties of the underlying chemical compounds. Several thousands such topological indices have been sug- gested,1 but only a dozen or so were proved to have true applicative power. One of these is the atom–bond connectivity (ABC) index. It was introduced in 1998 by Estrada et al.,2 but it attracted little attention. Only after the publication of a paper,3 ten years later, did the ABC index rapidly gain in popularity. It was shown2–4 that by means of the ABC index, it is possible to predict the thermo- dynamic properties of acyclic and cyclic saturated hydrocarbons, including those with large steric strain. Comparative studies5,6 confirmed that the ABC index yields significantly better results than other mathematically similar molecular structure descriptors. * Corresponding author. E-mail: furtula@kg.ac.rs # Serbian Chemical Society member. doi: 10.2298/JSC150901093F _________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2016 SCS. All rights reserved. 272 FURTULA, GUTMAN and DAS The ABC index is defined as follows. Let G be a molecular graph with n vertices, 1 2, , , nv v v . The edge of G, connecting the vertices iv and jv will be denoted by ( , )e i j . The degree id of the vertex iv is the number of the first neighbors of iv . Then: 2 i j i j i je d d ABC d d + − =  (1) with the summation going over all edges of the graph G. Nowadays, the theory of the ABC index is well developed, and its mathe- matical properties have been duly examined; for details see the recent papers7–13 and the references cited therein. Motivated by the success of the ABC index, Graovac and Ghorbani14 intro- duced its new variant, defined as: 1 2 GG 1 2 ( ) ( ) 2 ( ) ( ) + − =  ij ij ij ij ije n e n e ABC n e n e (2) In formula (2), 1( )ijn e is the number of vertices of G whose distance to the vertex iv is smaller than the distance to the vertex jv . Analogously, 2 ( )ijn e is the number of vertices of G whose distance to the vertex jv is smaller than to iv . Vertices equidistant from both iv and jv are ignored. More on the numbers 1( )ijn e and 2 ( )ijn e can be found elsewhere.15,16 This distance-based variant of the atom–bond connectivity index was until now studied only to a limited extent.14,17–19 Interestingly, none of the art- icles14,17–19 considered the simplest and most obvious question, namely: what is the relation between the original atom–bond connectivity index ABC, Eq. (1), and its modified version ABCGG, Eq. (2)? The aim of the present work is to provide an answer to this question. NUMERICAL WORK The ABC and ABCGG indices were calculated for several classes of isomeric alkanes and cycloalkanes. In all cases studied, it was found that between these two structure descriptors there is no (either linear or any other) correlation. A typical example is presented in Fig. 1. Not only that the two atom–bond connectivity indices are not correlated, but they also imply opposite ordering for structurally similar compounds. For instance, for 2-methylnonane, GG6.58 , 6.49= =ABC ABC (thus, GG>ABC ABC ), whereas for 3-methylnonane, ABC = = 6.47, GG 6.58=ABC (thus, GGABC ABC but GG≈ABC ABC ), whereas for 3,3-dimethyloctane, GG6.68 , 6.95= =ABC ABC (thus, < GGABC ABC ). Bearing in mind the good correlation properties of the original ABC index,2-5 it could be concluded that there is little hope that the ABCGG index would ever be found useful in chemical applications. _________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2016 SCS. All rights reserved. ON ATOM-BOND CONNECTIVITY MOLECULE STRUCTURE DESCRIPTORS 273 Fig. 1. Atom–bond connectivity indices of isomeric decanes plotted versus their ABCGG values. The data points lying above (resp. below) the line, satisfy GGABC ABC ). In the majority of cases (in this example: with only five exceptions), the ABC index is less than the ABCGG index. THE TWO ATOM–BOND CONNECTIVITY INDICES OF BENZENOID HYDROCARBONS In this section, a few basic properties of the two ABC indices of benzenoid molecules are established. For this, the facts from the well-elaborated topological theory of benzenoid hydrocarbons were used.20 An illustrative example is pro- vided in Fig. 2. Consider thus a benzenoid system with n vertices, m edges, h hexagons, ni internal vertices, and b bay regions on its perimeter.20 More details on parameter b can be found elsewhere.20–22 An edge is of ( , )r s -type, if it end-vertices have degrees r and s. A ben- zenoid system has only vertices of degrees two and three and therefore its edges are only of (2,2)- (2,3)- and (3,3)-type. Therefore, the term: ( 2) / ( )i j i jd d d d+ − in Eq. (1) is equal to 2/2 , 2/2 and 2/3 if the edge ije is of (2,2)-, (2,3) and (3,3)-type, respectively. This implies: _________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2016 SCS. All rights reserved. 274 FURTULA, GUTMAN and DAS Fig. 2. A benzenoid system with 11=h hexagons, 4=in internal vertices (indicated by heavy dots), three bays and one cove, and therefore with 3 1 1 2 5= × + × =b bay regions. This benzenoid sys- tem has 4 2 42= + − =in h n vertices and m = 5h + 1 – 52=in edges. According to Eq. (4), the number of (3,3)-edges is 33 11 1 4 5= − + + =m 19. 22 23 33 2 2 2 2 2 3 ABC m m m= + + where 22m , 23m and 33m stand for the number of edges of the (2,2)-, (2,3)- and (3,3)-type, respectively. Since 22 23 33m m m m+ + = : 33 33 2 2 ( ) 3 2 ABC m m m= + − (3) and 33m can be calculated by means of the identity: 33 1 im h n b= − + + (4) As 2/3 2/2< , it follows from Eq. (3) that the ABC index is bounded as: 2 2 3 2 m ABC m< ≤ (5) The equality on the right-hand side of (5) is attained only in the case of benzene ( 1h = ). In the case of benzenoid hydrocarbons, the quantities 1( )ijn e and 2 ( )ijn e , occurring in Eq. (2), can be calculated by the method of elementary edge- -cuts.15,23,24 An elementary edge-cut is a line segment that goes through the center of some hexagons, orthogonal to some edges, and intersects the perimeter exactly two times. The number of vertices lying on the two sides of an elementary edge- cut C is denoted by 1( )n C and 2 ( )n C . Since 1 2( ) ( )n C n C n+ = , only one among 1( )n C and 2 ( )n C requires evaluation, which is usually very simple. An illus- trative example is provided in Fig. 3. The number of edges intersected by the cut C will be denoted by ( )r C , cf. Fig. 3. If ije is any of the edges intersected by C, then: 1 1( ) ( )ijn e n C= and 2 2( ) ( )ijn e n C= For benzenoid hydrocarbons, this has the direct consequence that: 1 2 1 2 ( ) ( ) 2 ( ) ( ) ( )GG C n C n C ABC r C n C n C + − =  _________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2016 SCS. All rights reserved. ON ATOM-BOND CONNECTIVITY MOLECULE STRUCTURE DESCRIPTORS 275 holds, where the summation goes over all elementary edge-cuts. Since, in addition, for all elementary edge-cuts, 1 2( ) ( )n C n C n+ = , one finally obtains: 1 2 ( ) 2 ( ) ( ) GG C r C ABC n n C n C = −  (6) Fig. 3. Three elementary edge-cuts of the benzenoid system depicted in Fig. 2. Since this system has 42=n vertices, one has: 1 1 1 2 1( ) 2 , ( ) 3 , ( ) 42 3 39= = = − =r C n C n C , 2 1 2 2 2( ) 3 , ( ) 5 , ( ) 42 5 37= = = − =r C n C n C , and 3 1 3 2 3( ) 5 , ( ) 15 , ( ) 42 15 27= = = − =r C n C n C . On one side of any elementary edge-cut, there are at least three vertices and at most, one-half of the total number of vertices. Therefore, 13 ( ) / 2n C n≤ ≤ , which substituted back into Eq. (6), and bearing in mind that: ( ) C r C m= yields the bounds: GG 2 2 2 3( 3) − − ≤ ≤ − m n n ABC m n n (7) Equality on both sides of (7) is attained only in the case of benzene ( 1h = ). Combining the lower bound in (5) and the upper bound in (7), one obtains: GG 4( 3) 3( 2) − ≥ − ABC n ABC n in which equality is attained only in the case of benzene ( 6n = ). Thus, only in the case of benzene, do the two atom–bond connectivity indices coincide. For all other benzenoids, 2 , 10h n≥ ≥ and therefore: GG 4(10 3) 3(10 2) − > − ABC ABC i.e., GG7/6>ABC ABC where 7/6 1.0801= . If naphthalene is excluded, then 14n ≥ , and GG11/9>ABC ABC where 11 /9 1.1055= . Thus, in contrast to alkanes and cycloalkanes, the atom–bond connectivity index of any benzenoid hydrocarbon is always greater than the ABCGG index. Acknowledgments. The first author was partially supported by the Ministry of Education, Science and Technological Development of the Republic of Serbia, through Grant No. _________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2016 SCS. All rights reserved. 276 FURTULA, GUTMAN and DAS 174033. The third author was supported by the National Research Foundation funded by the Korean government by Grant No. 2013R1A1A2009341. И З В О Д МОЛЕКУЛСКИ СТРУКТУРНИ ДЕСКРИПТОРИ ПОВЕЗАНОСТИ ТИПА АТОМ–ВЕЗА БОРИС ФУРТУЛА1, ИВАН ГУТМАН1,2 и KINKAR CH. DAS3 1Природно–математички факултет Универзитета у Крагујевцу, 2Државни универзитет у Новом Пазару и 3Sungkyunkwan University, Suwon, South Korea Индекс повезаности типа атом–веза (ABC) је молекулски структурни дескриптор заснован на степенима чворова, чије су хемијске примене добро документоване. Године 2010. предложена је једна варијанта овог индекса (ABCGG) заснована на растојању. До сада, релације између ABC и ABCGG нису биле истраживане. У овом раду, установљене су основне карактеристике ове релације: индекси ABC и ABCGG нису корелисани, и оба случаја, ABC > ABCGG и ABC < ABCGG, се могу догодити код структурно сличних моле- кула. Међутим, у случају бензеноидних угљоводоника, ABC је увек већи од ABCGG. (Примљено 1. септембра, прихваћено 12. октобра 2015) REFERENCES 1. R. Todeschini, V. Consonni, Molecular Descriptors for Chemoinformatics, Wiley–VCH, Weinheim, 2009 2. E. Estrada, L. Torres, L. Rodríguez, I. Gutman, Indian J. Chem., A 37 (1998) 849 3. E. Estrada, Chem. Phys. 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Soc. 68 (2003) 549 17. K. C. Das, K. Xu, A. Graovac, Acta Chem. Slov. 60 (2013) 34 18. M. Rostami, M. Sohrabi-Haghighat, M. Ghorbani, Iran. J. Math. Chem. 4 (2013) 265 19. M. Rostami, M. Sohrabi-Haghighat, MATCH Commun. Math. Comput. Chem. 71 (2014) 21 20. I. Gutman, S. J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer, Berlin, 1989. 21. R. Cruz, I. Gutman, J. Rada, J. Serb. Chem. Soc. 78 (2013) 1351 22. I. Gutman, J. Serb. Chem. Soc. 79 (2014) 1515 23. I. Gutman, S. J. Cyvin, MATCH Commun. Math. Comput. Chem. 36 (1997) 177 24. A. A. Dobrynin, I. Gutman, S. Klavžar, P. Žigert, Acta Appl. Math. 72 (2002) 247 25. S. Klavžar, MATCH Commun. Math. Comput. Chem. 60 (2008) 255. _________________________________________________________________________________________________________________________Available on line at www.shd.org.rs/JSCS/ (CC) 2016 SCS. 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