Precise definitions are given for some classes of molecular graphs with one pentagon and otherwise hexagons: the monopentapolyhexes. The fluoranthenoid and fluorenoid systems belong to monopentapolyhexes. Complete mathematical solutions, using combinatorial summations on the one hand and generating functions on the other hand, are given for the numbers of catacondensed simply connected monopentapolyhexes (catafluorenoids and the corresponding helicenic systems). Generating functions and numerical values are included

Brunvoll, J.

Cyvin, B. N.

Cyvin, S. J.

Guo, X. F.

Tosic, R.

Zhang, F. J.

郭晓峰

Journal of Molecular Structure THEOCHEM

Xiamen University Institutional Repository

Journal of Molecular Structure (Theochem), 285 (1993) 179- 185 0166-1280/93/%06.00 0 1993 - Elsevier Science Publishers B.V. All rights reserved 179 Graph-theoretical studies on fluoranthenoids and fluorenoids: enumeration of some catacondensed systems” S.J. Cyvin , . . *9a B N Cyvin” J Brunvolla, Zhang Fujib, Guo Xiaofengb, R. To%? , . aDivision of Physical Chemistry, The University of Trondheim, N-7034 Trondheim-NTH, Norway ‘Department of Mathematics, Xinjiang University, Wulumuqi, Xinjiang 830046, People’s Republic of China ‘Institute of Mathematics, Faculty of Science, University of Novi Sad, YU-21000 Novi Sad, Yugoslavia (Received 11 December 1992; accepted 5 January 1993) Abstract Precise definitions are given for some classes of molecular graphs with one pentagon and otherwise hexagons: the monopentapolyhexes. The fluoranthenoid and fluorenoid systems belong to monopentapolybexes. Complete mathematical solutions, using combinatorial summations on the one hand and generating functions on the other hand, are given for the numbers of catacondensed simply connected monopentapolyhexes (catafluorenoids and the corresponding helicenic systems). Generating functions and numerical values are included. Introduction Fluoranthenoids (or fluoranthenes) and fluore- noids (or fluorenes) are polycyclic hydrocarbons consisting of one five-membered ring and varying numbers of six-membered rings [l]. These two classes are distinguished by an even or odd number of carbon atoms (and hydrogen atoms), respectively, in the pertinent hydrocarbons. Among the fluorenoids are the important cata- condensed hydrocarbons without internal carbon atoms, i.e. in which there is no carbon atom shared by three rings. Catacondensed fluorenoids have the formulae C4r+ lH2r+3, where r is used to designate the number of rings. These compounds are radicals and as such chemically unstable. However, several of the corresponding C4 r + 1 H2 r + 4 molecules are known, such as cyclopentadiene C5H6 * Corresponding author. 0 This project was supported by NSFC. corresponding to cyclopentadienyl CsH5. Similarly, the molecules corresponding to the following radicals are current in the chemical literature (and laboratories): indenyl CsH,, fluorenyl Ct3H9, benzo[a]fluorenyl (chrysofluorenyl) C17H119 and two isomers of C2tHt3, i.e. dibenzo[a, g]fluorenyl and dibenzo[b, hlfluorenyl. Dias [2] has identified the three isomers of Ct3H9 fluorenoids, but otherwise the numbers of cata- condensed fluorenoid isomers seem not to be known. In the present work we give a complete mathematical solution for the numbers of C4r+lH2r+3 fluorenoid isomers augmented by the corresponding helicenic systems. Fluorenoid (and fluoranthenoid) systems Fluorenoid systems (or simply fluorenoids), which have their chemical counterparts in fluorenoid hydrocarbons, belong to a class we shall refer to as monopentapolyhexes. This class 180 S.J. Cyvin et al./J. Mol. Struct. (Theochem) 285 (1993) 179-185 Fig. 1. The monopentahexagonal lattice, where the penta- gon is marked by a circle. The perimeter of benzo[u]fluorenyl (chrysofluorenyl) is indicated in bold. is similar to polyhexes, which are represented exclusively by congruent regular hexagons in a plane. The designation “monopentapolyhex” is open for extension to analogical terms like dipenta- polyhex, polypentapolyhex, monooctapolyhex, etc. A monopentahexagonal lattice is similar to the hexagonal lattice used in the definition of polyhexes and benzenoids [3,4] in particular. The monopentahexagonal lattice is a planar array of hexagons built around one regular pentagon. Only each set of hexagons at the same distance from the pentagon are mutually congruent, but none of the hexagons can be regular. The fluorenoid (and fluoranthenoid) systems are simply connected geometrically planar (non- helicenic) monopentapolyhexes. As such they are analogous to benzenoids (when the latter class is defined as simply connected geometrically planar polyhexes). A fluorenoid (or fluoranthenoid) is defined by a cycle on the monopentahexagonal lattice and its interior, which should include the pentagon. The cycle represents the boundary of the fluorenoid/fluoranthenoid system in question and is usually called the perimeter. An illustration is provided in Fig. 1. The subject of the present work are the catacon- densed simply connected monopentapolyhex systems, which may be either geometrically planar (nonhelicenic) or geometrically nonplanar (helicenic). These subclasses shall be referred to as catafluorenoids and catahelifluorenes, respectively. A more detailed scheme of classification of the simply connected monopentapolyhexes is proposed in Fig. 2. For the sake of comparison the r I catafluorenoid] fluorenoid/ fluoranthenoid perifluorenoid/perifluoranthenoid monopentapolyhex helifluoranthene perihelifluorene/perihelifluoranthene benzenoid simply connected polyhex (fusene) catahel i cene perihelicene Fig. 2. Classification schemes of simply connected monopentapolyhexes and simply connected polyhexes. S.J. Cyvin et al./J. Mol. Struct. (Theochem) 285 (1993) 179-185 181 Table 1 Numerical values of some variables defined in the texta h uh vh FI +A0 4 + h @) Fl+h 1 1 1 1 0 1 2 3 1 2 1 3 3 10 2 6 3 9 4 36 2 19 16 35 5 137 5 71 66 137 6 543 5 274 300 574 I 2219 15 1117 1314 2431 8 9285 15 4650 5884 10534 9 39587 51 19819 26304 46123 10 171369 51 85710 118633 204343 11 751236 188 375712 537255 912967 12 3328218 188 1664203 2447172 4111375 a See Eqs. (2), (13) and (17)-(19). corresponding classification scheme for the simply connected polyhexes [5,6] is included in Fig. 2. Enumeration of the catacondensed simply connected monopentapolyhexes by combinatorial summations General Let the title class be designated by F. A general formula has been derived for the numbers of nonisomorphic systems of the class F as a function of h, where r=l+h (1) and h is used to designate the number of hexagons, so that r is the total number of polygons (one pentagon and the rest hexagons, if any). A member of the class F is either one pentagon (cyclopentadienyl), or one pentagon with one or two annelated catacondensed simply connected polyhexes (catafusenes). The method which is employed here is referred to as combinatorial summations. It has previously been applied to catafusenes with regular trigonal symmetry [7] and to catafusenes in general [8]. The catafusenes rooted by an edge play a crucial role in the method. The catafusenes which are annelated to the pentagon are only annelated by the root edges. Rooted catafusenes The number of edge-rooted catafusenes with h hexagons, say U,,, is given recursively by [7-91 u, = 1, lJ, = 3ui = 3, u,,, = 3Uh h-l + c utuh-i i=l (h = 2,3,4, . . .) (2) By definition, us = 0 (3) Numerical values of uh for 1 < h < 12 are found in Table 1. Annelation schemes A system of the class F may have (i) one catafusene or (ii) two catafusenes annelated to the pentagon, apart from being the pentagon itself. Examples for the schemes (i) and (ii) are found in Fig. 3. In each of these cases only one scheme is possible when isomorphic systems are to be avoided. Let the numbers of nonisomorphic systems with r polygons each be designated by F,(i) 182 S.J. Cyvin et a/./J. Mol. Struct. (Theochem) 285 (1993) 179-185 Fig. 3. Examples of the schemes of annelations to a pentagon (marked by a circle): (i) one annelated catafusene; (ii) two annelated catafusenes. ( > ii and F,(ii) under schemes (i) and (ii), respectively (see Fig. 3). Then the total number F, is F, = e(i) + I;, (ii) (ha 1) (4) One appendage (scheme (i)) In this treatment, h 2 1. The systems under consideration are split into F, (i) = F,” (i) + Fp (i) (5) where M and A refer to mirror-symmetrical (CT,) and unsymmetrical ( CS) systems, respectively. Here IU-I)/21 FEh(i) = _. 2 A(1 + Vi) i=O (6) where the floor function is employed in the upper index: [xJ is the largest integer not larger than x. Now U,, will count the unsymmetrical systems twice, but the mirror-symmetrical only once, so that iJ, = FEh(i) + 2FP+h(i) A combination of Eqs. (5)-(7) gives (7) I+ 11121 c (‘+Ui) i=O I (8) Two appendages (scheme (ii) Here it is assumed ha 2. We write, by analogy with Eq. (5) F,(ii) = K”(ii) + Fp(ii) (9) where F$(ii) = eU,,/2 (10) here E = 1 when h is even, and e = 0 when h is odd. In the latter case the symbol U,i2 is meaningless, but it is understood that the whole term vanishes. In the present case it is found h-l c uiuh-i = Fzh(ii) + 2F&h(ii) i=l which gives, together with Eqs. (9) and (10) Fi +h(ii) = i h-l EUh/2 + c uiuh-i i=l (11) (12) Simplification of formulae In the first place, introduce the simplified notation l(h - 1)N vh = C (1 + ui) i=O (hall (13) for the summation of Eq. (8). The numbers of Eq. (13) are most easily found according to the initial condition and recurrence relation v, = 1, vh+, = vh + eU,,,2 (14) where E is defined after Eq. (10). Numerical values of vh for 1 <h < 12 are included in Table 1. Now we are ready for the following simplifi- cations. First, the simplified notation of Eq. (13) is introduced in Eq. (8). Next, the two terms in S.J. Cyvin et al./J. Mol. Struct. (Theochem) 285 (1993) 179-185 183 parentheses in Eq. (12) may be substituted by fuhf2 = vh,, - vh (h> 1) (15) and the class F as Fi+h =;(Uh+i + vh+l) - uh (19) For numerical values of Fl fh (1 <h < 12), see Table 1. The above equation is also valid for h = 0, in which case it gives F, = 1. This accounts for the pentagon alone. h-l c uiuh_i = uh+, - 3Uh (ha2) (16) i=l as obtained from Eqs. (14) and (2), respectively. The results are Fl+h(i) ‘$c”h + vh) (17) Fl+h(ii) =$(Uh+l + vh+l - 3uh - vh) (18) Both Eqs. (17) and (18) are valid for hal. Numerical values of the variables in these equations are again found in Table 1. Total numbers with distributions into symmetry groups On inserting the expressions from Eqs. (17) and (18) into Eq. (4) one obtains the final result for the number of nonisomorphic systems of Table 2 From the above analysis we are also able to give the numbers of nonisomorphic systems of the class F separately for the mirror-symmetrical and unsymmetrical (C,) systems, say FrM and FrA, respectively. They are F% = vh+l (20) F&h = i(“h+l - Vh+l) - uh (21) Here the numbers of Eq. (20) include Fy = 1 for the pentagon itself. It is the only system of the class F with the regular pentagonal (D& symmetry; otherwise the mirror-symmetrical systems of this class all belong to C,,. In Table 2 the numbers of the monopentapolyhexes of the class F are given, including their distributions into the symmetry Numbers of catacondensed simply connected monopentapolyhexes (catafluorenoids + catahelifluorenes); the symmetry distribution is included r Dsh C2” cs Total (F,) 1 1 0 0 1 2 0 1 0 1 3 0 2 1 3 4 0 2 7 9 5 0 5 30 35 6 0 5 132 137 7 0 1.5 559 574 8 0 15 2416 243 1 9 0 51 10483 10534 10 0 51 46072 46123 11 0 188 204155 204343 12 0 188 912779 912967 13 0 731 4110644 4111375 14 0 731 18636572 18637303 15 0 2950 84985825 84988775 16 0 2950 389586145 389589095 17 0 12235 1794268460 1794280695 18 0 12235 8298524480 8298536715 19 0 51822 38527095859 38527147681 20 0 51822 179487051589 179487103411 184 S.J. Cyvin et al./J. Mol. Struct. (Theochem) 285 (1993) 179-185 groups of interest, i.e. &, C,, and C,. The totals (F,) in Table 2 are found to be consistent with those of Table 1 (F,+h). Enumeration of the catacondensed simply connected monopentapolyhexes by generating functions Generating functions provide an elegant way to represent different sets of numbers. Such functions were also derived for the main numbers of the present study (including 6). In the celebrated work of Harary and Read [9], the enumeration problem for the catacondensed simply connected polyhexes (catafusenes) was solved in terms of generating functions. The same procedure can easily be adapted to the class F of the present study, for which the enumeration problem is in fact substantially simpler. There- fore, it should not be necessary to treat this pro- cedure in detail here. We shall only quite briefly translate some of the relations already given above into the language of generating functions. The generating function for the numbers of edge- rooted catafusenes ( Uh) is known [8,9] and reads U(x) = 2 Qx’ = ix-‘[l - 3x - (1 - X)“2 i=l x (1 - 5X)“2] (22) From Eq. (15) we find 1 + U(X2) = x-l V(x) - V(x) (23) where V(x) is the generating function for V,. It follows that V(x) = x(1 - x)-‘[1 + U(x2)] (24) and consequently V(x) = -p v,xi = &‘[l +x - (1 - x>-’ i=l x (1 - x2)1’2(1 - 5x2)“2] (25) The main result, i.e. Eq. (19), is transferred to a relation between generating functions like x-IF(x) = &‘[U(x) + V(x)] - U(x) (26) Hence F(x) = 1 [( 1 - 2x) U(x) + V(x)] (27) On inserting the functions from Eqs. (22) and (25) it was obtained as F(x)=~P’~x’=~x-‘[~-~~+~x~-; i=l x (1 - 2x)( 1 - x)1’2( 1 - 5x)“2 -I(1 - x)-‘(1 -~~)“~(l - 5~~)“~] (28) Introduce also the generating functions for the separate numbers which pertain to the different symmetries, say FM(x) and F*(x), which pertain to I;;” and F,*, respectively. Then the following relations, which are consistent with Eqs. (20) and (21), are valid. FM(x) = V(x) (29) where V(x) is given by Eq. (25), and F*(x) = ;[(l - 2x)U(x) - V(X)] (30) which again could be given explicitly by inserting from Eqs. (22) and (25). In the present work a complete mathematical solution was achieved for the numbers of monopentapolyhexes of the class F: catacon- densed simply connected monopentapolyhexes (catafluorenoids + catahelifluorenes). The method of combinatorial summations was employed, but the solution is also given in terms of generating functions. It was checked numerically that the two representations, e.g. Eqs. (19) and (28), are consistent. In fact, Table 1 was produced entirely from the combinatorial summations, without use of computers. The same is the case for the numbers at r< 13 in Table 2. Next, a simple computer program was implemented for the generating functions. This resulted in data which were completely consistent (as they should be) with the r < 13 numbers of Table 2 and made an extension of Table 2 practicable. S.J. Cyvin et al./J. Mol. Struct. (Theochem) 285 (1993) 179-185 185 Whereas an immense body of literature has References accumulated on the graph-theoretical studies of polyhexes, very little has previously been done for 1 monopentapolyhexes in this area. The proposed terms like dipentapolyhexes, polypentapolyhexes, 2 monooctapolyhexes, etc. suggest that many 3 related nonbenzenoid systems, possibly with 4 generalizations, could be treated according to the 5 present approach. 6 Acknowledgment 7 Financial support to B.N.C. from the 8 Norwegian Research Council for Science and the 9 Humanities is gratefully acknowledged. J.R. Dias, Polycyclic isomers and heteroatom analogs of benzenoid hydrocarbons, Handbook of Polycyclic Hydrocarbons, Part B, Elsevier, Amsterdam, 1988. J.R. Dias, Chem. Phys. Len., 185 (1991) 10. I. Gutman, Bull. Sot. Chim. Beograd, 47 (1982) 453. I. Gutman and S.J. Cyvin, Introduction to the Theory of Benzenoid Hydrocarbons, Springer, Berlin, 1989. J. Brunvoll, S.J. Cyvin and B.N. Cyvin, J. Mol. Struct. (Theochem), 235 (1991) 147. B.N. Cyvin, J. Bnmvoll and S.J. Cyvin, Top. Curr. Chem., 162 (1992) 65. S.J. Cyvin and J. Brunvoll, Polycyclic Aromat. Compd, 2 (1991) 209. S.J. Cyvin and J. Brunvoll, J. Math. Chem., 9 (1992) 33. F. Harary and R.C. Read, Proc. Edinburgh Math. Sot. Ser. II, 17 (1970) 1. 

GRAPH-THEORETICAL STUDIES ON FLUORANTHENOIDS AND FLUORENOIDS - ENUMERATION OF SOME CATACONDENSED SYSTEMS

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GRAPH-THEORETICAL STUDIES ON FLUORANTHENOIDS AND FLUORENOIDS - ENUMERATION OF SOME CATACONDENSED SYSTEMS

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