Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons

Transcription

1 /;C' // - MODELS IN CHEMISTRY 136 (5-6), pp (1999) Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons IVANGUTMAN1, ELENA V. KONSTANTINOVA2 and VLADIMIR A. SKOROBOGATOV2 1 Faculty of Science. University of Kragujevac, P.O. Box 60, YU Kragujevac, Yugoslavia 2Sobolev Institute of Mathematics, Russian Academy of Sciences, Siberian Branch, Novosibirsk , Russia Received March 12, 1999 A new approach to the Clar aromatic sextet theory of benzenoid hydrocarbons is put forward, in which the circles in the Clar structural formulas are viewed as hyperedges of the respective molecular hypergraph. The distance-based topological indices of the graph- and hypergraph-representation of a benzenoid hydrocarbon differ significantly. The basic rules governing these differences are established. Introduction In the majority of organic molecules covalent chemical bonds exist between distinct pairs of atoms. The bonding in such a molecule is then described by a single classical structural formula. The natural mathematical model of the respective molecular structure is the molecular graph [1,2]. There are, however, quite a few examples of molecules in which several (more than two) atoms are covalently bound by means of a chemical bond, a so-called polycentric bond. Most often polycentric bonds are encountered in inorganic clusters, organometallic (sandwich-type) compounds, etc. As will be explained below, bonds of the same kind occur also in polycyclic conjugated molecules, benzenoid systems in particular. Two of the present authors introduced recently [3, 4] the concept of molecular hypergraph, as the mathematical tool suitable for modeling the structure of molecules with polycentric bonds. For the following considerations it is purposeful to recall the definitions of graphs and hypergraphs [1, 2, 5]. 1: /99/ $ Akadmial Kiadd, Budapest

2 540 GUTMAN et al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons Molecular hypergraphs Let V={v[, v2,..., v,,} be a set consisting of n, n>l, unspecified elements, called vertices. Let E={E\,E2,...,Em} be a family of some distinct subsets of V. It is required that each }, /= 1,2,...,m, contains at least two elements, i.e., embraces at least two vertices. The elements of E are called hyperedges. The hyperedges embracing exactly two vertices will be referred to as simple edges, whereas those embracing three or more vertices as proper hyperedges. A hypergraph H = (V,E) is a mathematical object consisting of vertices and hyperedges; the sets of vertices and hyperedges of H are the above described V= V(H) and E = E(H), respectively. If,-= {v/,, v/,,...,vih} is a hyperedge of H, then we say that the vertices v,-,, v,-,,..., v/;i are mutually adjacent, i.e., that the hyperedge E-t connects the vertices v/,, v,-,,..., v/a. A graph G = (V,E) is a special case of a hypergraph in which all hyperedges are simple, i.e., all (hyper)edges connect just two vertices. Clearly, if vertices represent the atoms of a molecule, then every simple edge represents a regular, two-centric covalent chemical bond. If polycentric bonds exist, these are naturally modeled by means of hyperedges. Consequently, the structure of molecules with only two-centric covalent bonds corresponds in a natural way to molecular graphs, whereas the structure of molecules possessing polycentric covalent bonds is best represented by means of molecular hypergraphs; examples of molecular hypergraphs are found later in the text; for further details and additional examples see [3,4]. Clar theory It is one of the standard results of theoretical organic chemistry that benzenoid hydrocarbons (and polycyclic conjugated molecules in general) have no unique classical structural formula. The conjugation modes of their 7t-electrons must be visualized by means of several so-called Kekule structures [6], none of which correctly describes the actual situation. An attempt to overcome this century-long problem was proposed by Clar in the 1950's [7, 8] and eventually elaborated in due detail in his seminal book [9]. According to Clar, the electronic structure of benzenoid hydrocarbons is represented by, what nowadays is known as, Clar aromatic sextet formulas (CASFs). These formulas contain circles drawn inside some of the hexagons of the respective molecule. The formal rules for constructing a CASF are the following [6, 10]: - c u - MODELS IN CHEMISTRY

3 GUTMAN et al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons 541 a) circles must not be drawn into adjacent hexagons, b) the part of the carbon-atom skeleton, not covered by circles, must possess a Kekule structure, c) obeying conditions a) and b), as many circles as possible must be drawn. The circles in a CASF symbolize aromatic sextets, groups of six n-electrons, which - according to the Clar theory - are dominantly located in the respective hexagons of the benzenoid molecule. The aromatic sextets are indicators of a strong stabilizing effect caused by cyclic conjugation, and imply (local) aromaticity of the respective domains of the benzenoid molecule. This is usually in good agreement with experimental findings and has been corroborated by several theoretical considerations [11-15]. The Clar aromatic sextet, located inside a given hexagon, can be understood as a six-centric chemical bond connecting the six carbon atoms that form the respective hexagon. By means of Clar theory many benzenoid molecules gain a unique structural formula. In what follows we will be concerned only with such benzenoid systems. In Fig. 1 are shown two examples of this kind, namely the CASFs of phenanthrene and dibenzo \fg,o/?]naphthacene. la Fig. 1. Clar aromatic sextet formulas of phenanthrene (la) and dibenzo[/g,op]naphtnacene (Ila); diagrams Ib and lib show the numbering of the vertices in the respective molecular graphs and hypergraphs. For details see text C - MODELS IN CHEMISm

4 542 GUTMAN et al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons Details of the Clar theory can be found in the books [6, 9]; for quantitative studies of the relations between Clar theory and cyclic conjugation in benzenoid and non-benzenoid polycyclic conjugated molecules see [15-18] and the references quoted therein. Distance-based structure-descriptors The (topological) distance between two vertices of a graph is defined as the number of edges in a shortest path connecting these vertices. In full analogy to this, we understand that the distance between two vertices of a hypergraph is the number of hyperedges in a shortest path connecting these vertices. (For examples see the subsequent section.) The distance between the vertices v(- and v- in the graph G and hypergraph H will be denoted by d(vf, vy-1 G) and d(vt, Vj\H), respectively. In the case of molecular graphs several vertex-distance-based graph invariants found chemical applications and have been extensively studied in the chemical literature. These are usually referred to as topological indices, 77s [1, 2]. Of the plethora of such 77s we have focused our attention to the following three: the Wiener index, W, the mean square distance index, SDI and the vertex distance index, VDI. This choice was made because the same TIs were examined also in our previous works [3, 4]. Molecular graphs are necessarily connected. Therefore the distance d(vf,vj\g) is well-defined for all pairs of vertices (v,-, v-). Denote the number of vertex pairs of G at distance k by d(g,k). If G has n vertices, then because there is a total of ( ) vertex pairs in G. Now, in terms of the quantities d(g,k) the Wiener index, the mean square distance index and the vertex distance index of the molecular graph G are defined as (1) &>/<«= T^- (2) I' '*>! - MODELS IN CHEMISTRY

5 GUTMAN el al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons 543 and VW(G)=!</(G,*)2, (3) *>l respectively. The Wiener index is the oldest and one of the most thoroughly studied molecular structure-descriptors (for review see [19-21]). The indices SDI and VDI, proposed by Rouvray [22], reflect structural features other than W. For instance, whereas Wis related to the average vertex distance, d =2M(2>, the variance of the vertex distances is given by (SDf)2-(d)2 and is thus related to the 5D/-index. Whereas both W and SDI decrease with increasing branching of the molecular skeleton, the VD/-index exhibits an opposite, decreasing trend. The Wiener index, the mean square distance index and the vertex distance index of the molecular hypergraph H are defined analogously: in Eqs (l)-(3) one only has to everywhere exchange the symbols G by H. The Clar hypergraph According to what has been outlined in the preceding section, it should be clear that the Clar aromatic sextet formulas may be understood as hypergraphs, in which the aromatic sextet is just a hyperedge embracing the six vertices lying on the respective hexagon. We illustrate this observation by the two benzenoid systems from Fig. 1. In the usual graph-representation, the molecular graph of phenanthrene has 14 vertices and 16 edges, cf. diagram Ib in Fig. 1. The molecular hypergraph of phenanthrene has the same 14 vertices, but possesses 18 hyperedges - 16 simple edges: {1,2}, {2,3}, {3,4}, {4,5}, {5,6}, {1,6}, {1,7}, {7,8}, {8,9}, {9,10}, {10,11}, {11,12}, {7,12}, {2,13}, {8,14} and {13,14}, as well as two proper hyperedges: {1,2,3,4,5,6} and {7,8,9,10,11,12}. Similarly, the molecular graph of dibenzo[/g, op}]naphthacene has 24 vertices and 29 edges (cf. lib) whereas the molecular hypergraph of the same hydrocarbon has 33 hyperedges - 29 simple edges as well as four proper hyperedges: {1,2,3,4,5,6}, {7,8,9,10,11,12}, {13,14,15,16,17,18} and {19,20,21,22,23,24}. Because the above described molecular hypergraphs emerged from Clar theory we find it justified to name them Clar hypergraphs. Thus, every CASF may be viewed as a hypergraph. '.u

6 544 GUTMANet al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons At this point one should ask what is the difference (if any) between a standard interpretation of a CASF and its interpretation as a hypergraph. The fundamental difference is that the (topological) distances between vertices have changed significantly. For instance, in the molecular graph of phenanthrene the distance between the vertices 4 and 10 is 7, because each shortest path connecting these vertices contains seven edges (cf. Fig. 1). There are 5 such shortest paths, one of which is {4,5}, {5,6}, {6,1}, {1,7}, {7,8}, {8,9}, {9,10} In the Clar hypergraph the distance between the vertices 4 and 10 is only 3 because the shortest path (which is unique) connecting these vertices consists of three hyperedges: {1,2,3,4,5,6}, {1,7}, {7,8,9,10,11,12} It is easy to see that the following general regularity is obeyed: Rule 1. If G is the molecular graph of a benzenoid system and H the respective Clar hypergraph, then rf(v(-, v-\h) < d(vf, v- G) holds for any two vertices v(-, v-. Bearing in mind the above, we have undertaken studies of distance-based TIs of benzenoid systems, comparing the values pertaining to Clar hypergraphs with those for molecular graphs. The results obtained are summarized in the subsequent section. Relations between distance-based structure-descriptors of benzenoid graphs and Clar hypergraphs Taking into account Rule 1 and the basic properties of the topological indices W; SDIand VDI (see Eqs. (l)-(3)), we immediately arrive at Rule 2. If G is the molecular graph of a benzenoid system and H the respective Clar hypergraph, then W(G) > W(H), SDI(G) > SDI(ff), and VDI(G) < VDI(H). In what follows we denote by AW, ASD/ and AVDI the differences W(G)-W(H), SDI(G)-SDI(H) and VDI(H)-VDI(G), respectively. According to Rule 2 all three differences are necessarily positive-valued. The next rule was obtained by calculating W(G), W(H), SDI(G), SDI(H), VD1(G) and VDI(H) for a large number of benzenoid hydrocarbons having a unique CASF. Rule 3. For 77= W, SDI, VDI there is a very good linear correlation between and TI(G). TI(H)

7 GUTMAN et al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons 545 In particular, we obtained the following. Recall that benzenoid systems in which there are no double bonds in the CASF, i.e., in which all n-electrons belong to aromatic sextets, are called fully-benzenoid [6]. For the class of the 28 fully-benzenoid molecules with 11 or less hexagons (whose structures are given in [23]) we found: W(H) = (0.649 ± 0.006) W(G) + bw; r = , SDI(H) = (0.67 ± 0.02)SZ)/(G) + bsdl; r = 0.992, VDI(H) = (1.97 ± 0.04) VDI(G) + bvd{; r = 0.995, where r stands for the correlation coefficient. The calculated values of the parameters bw, bsdl and bvdi are 4 ± 20, -0.2 ± 0.1 and 400 ± 2300, respectively, and therefore it is reasonable to set them equal to zero in the above expressions. For the class of the 16 benzenoid systems with two- or three-aromatic sextets (=hyperedges), which are not fully-benzenoid, but which have a unique CASF we found W(H) = (0.72 ± 0.01) W(G) + bw; r = 0.998, SDI(ff) = ( )S >/(G) + bsd[; r = 0.96, VDI(ff) = (1.54 ± 0.03) VDI(G) + bvd,\ = 0.997, where bw=-3q±\0, bsdi = and bvdi= 1000±300, which should be neglected as well. Rule 4. The values of AW, AS >7 and AVD/ rapidly increase with the increasing size (= number of vertices and/or number of aromatic sextets) of the benzenoid molecule and vary to a much lesser extent among benzenoid isomers. Rule 4 is illustrated by the data given in Fig. 2. As another illustration of Rule 4 we give the formulas for W(G), W(H) and AW for the homologous series whose first members are the fully-benzenoid molecules III (p=0), IV (p= 1), V (p=2) and X (p=3), see Fig. 2: W(G) = 96p3 +216p p-45 for p > 1, W(ff) = 48p p2 + 78p + 15 for p > 0, A1V = 48p3 + 24p p + 60 for p > 1. We see that AW increases as the third power of the number of aromatic sextets ( = hyperedges).

8 546 GUTMAN et al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons V VI VII VIII Fig. 2. Examples illustrating the size-dependency of AW, ASD/ and AVD/ and their relatively low variation in the case of isomers (V-IX); for the molecules HI-X, the AW-values are: 180, 756, 1956, 1884, 1740, 1812, 1668 and 4068, respectively, the ASDAvalues are 1.35, 2.03, , 2.32, 2.49, 2.22 and 3.38, respectively, the AVX>/-values are 4770, 24004, 70952, 69870, 66176, 68320, and , respectively In a sharp contrast to Rule 4 we find that the A-values are almost independent of the number of hexagons. XI XII XIII XIV Fig. 3. Examples illustrating the almost-independence of AW, &SDI and AVD/ of the number of hexagons, provided the number of aromatic sextets is fixed; for the molecules XI-XIV, the AW-values are: 1524, 1508, 1492 and 1476, respectively, the A5Z)/-values are 2.00, 1.99, 1.97 and 1.96, respectively, the AVDAvalues are 96372, 88116, and 85824, respectively -MODELS IN CHEMISTRY

9 GUTMANet al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons 547 Rule 5. For benzenoid molecules with a fixed number of aromatic sextets, but with different number of hexagons, the values of AW, ASD/and AVD/are nearly constant. Rule 5 is illustrated by the data given in Fig. 3. Concluding remarks The Clar aromatic sextet theory was introduced without a sound theoretical foundation and without any mathematical formalism. Its remarkable success in describing, rationalizing and predicting experimental facts is not fully understood. By pointing out a hitherto overlooked..hypergraph connection" we hope to make a step forward towards constructing (or reconstructing) the mathematics of Clar theory. The main goal of this paper is to call the attention of the chemical community to a novel approach towards the understanding of the mathematical contents of the Clar aromatic sextet formulas: It is possible to view them as hypergraphs. The Clar hypergraph, defined in this paper, is not just another equivalent way of describing the bonding in polycyclic aromatic ^-electron systems. We showed that the Clar hypergraph has properties quite different from the traditional molecular graph. These differences are best manifested in the case of distance-based structuredescriptors. A few rules (sometimes semiquantitative, but always generally valid) are established for them. These rules pinpoint the main structural features of benzenoid molecules, reflected (or not reflected) in the differences between the topological indices of the molecular graphs and hypergraphs. Finer structure-effects remain still to be elucidated. The considerations in this work were restricted to benzenoid molecules with a unique Clar aromatic sextet formula. It must be noted, however, that there are cases when the n-electron conjugation in the benzenoid molecule needs to be described by several such formulas, i.e., there are benzenoid systems for which the CASF is not unique. The properties of the hypergraph-representations of these molecules, which certainly are less straightforward and more perplexed, will be the subject of a later study. References [1] GUTMAN. I., POLANSKY, O. E.: Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin [2] TRINAJST1C. N.: Chemical Graph Theory. CRC Press. Boca Raton [3] KONSTANT1NOVA. E. V.. SKOROBOGATOV, V. A.: Vychislitel'nye Sistemy (Novosibirsk), 151, 55 (1994) [4] KONSTANTINOVA, E. v.. SKOROBOGATOV, v. A..- j. Chem. inf. Comput. Sci., 35,472 (1995) (",, - MODELS IN CHEMISTRY

10 548 GUTMANet al.: Molecular hypergraphs and Clar structural formulas of benzenoid hydrocarbons [5] HARARY, F.: Graph Theory, Addison-Wesley, Reading, 1969 [6] GlTTMAN, I., CYVIN, S. J.: Introduction to the Theory of Benzenoid Hydrocarbons, Springer- Verlag, Berlin, 1989 [7] CLAR, E., ZANDER, M.: /. Chem. Soc (1958) [8] CLAR, E., IRONSIDE, C. T.t ZANDER, M.: /. Chem. Soc., 142 (1959) [9] CLAR, E.: The Aromatic Sextet, Wiley, London, 1972 [10] Gl/TMAN, I.: /. Serb. Chem. Soc., 47, 453 (1982) [11] PAUNCZ, R., COHEN, A.: J. Chem. Soc., 3288 (1960) [12] POLANSKY, 0. E., DERFUNGER, G.: Int. J. Quantum Chem., 1, 379 (1967) [13] A1HARA, J.: Bull. Chem. Soc. Japan, 49, 1429 (1976) [14] AIDA, M., HOSOYA, H.: Tetrahedron, 36, 1317 (1980) [15] ZHY, H., JIANG, Y.: Chem. Phys. Lett., 193, 446 (1992) [16] GUTMAN, I., PETROVIC, V.: Monatsh. Chem., 126, 1179 (1995) [17] GLTTMAN, I., IVANOV-PETROVIC, V.: /. Mol. Struct. (THEOCHEM), 389, 227 (1997) [18] GUTMAN, I.: J. Mol. Struct. (THEOCHEM-), 428, 241 (1998) [19] GUTMAN, I., YEH, Y. N.. LEE, S. L., LUO, Y. L.: Indian J. Chem., 32A. 651 (1993) [20] NIKOUC S., TRINAJSnC, N., MlHALIC, Z.: Croat. Chem. Acta, 68, 105 (1995) [21] LUKOVITS, I.: Kemiai Kodemenyek, 82, 107 (1996) [22] ROUVRAY, D. H.: In: Chemical Applications of Topology and Graph Tlieory, (Ed. KING R. B.), Elsevier, Amsterdam, 1983, pp [23] CYVIN, B. N., BRUNVOLL, J., CYVIN, S. J., GUTMAN, L: Commun. Math. Chem. (MATCH}, 23, 163 (1988) C a - Mooas IN CHEMISTRY