A polyene graph is a tree that can be embedded in a hexagonal lattice.
Systems of polyene graphs attached to one hexagon are considered. Overlapping edges and/or vertices (geometrically nonplanar systems) are allowed. A complete mathematical solution is presented in terms of a generating function for the numbers of isomers of the systems in question. The corresponding geometrically planar systems, referred to as styrenoids, are enumerated by computer programming. Finally, in the Appendix, the generating function is given for the numbers of free polyene graphs

B. N. Cyvin

E. Brendsdal

J. Brunvoll

S. J. Cyvin

English

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CROATICA CHEMICA ACTA CCACAA68 (1) 63-73 (1995)ISSN 0011-1643CCA-2212 Original Scientific PaperIsomers of Polyenes Attached to BenzeneB. N. Cyvin, E. Brendsdal, J. Brunvoll and S. J. CyvinDepartment of Physical Chemistry, The University of Trondheim,N-7034 Trondheim-NTH, NorwayReceived June 6, 1994; revised October 10, 1994; accepted November 24, 1994A polyene graph is a tree that can be embedded in a hexagonal lattice.Systems of polyene graphs attached to one hexagon are considered. Over-lapping edges and/or vertices (geometrically nonplanar systems) are al-lowed. A complete mathematical solution is presented in terms of a gener-ating function for the numbers of isomers of the systems in question. Thecorresponding geometrically planar systems, referred to as styrenoids, areenumerated by computer programming. Finally, in the Appendix, the gen-erating function is given for the numbers of free polyene graphs.INTRODUCTIONThe CnHn + 2polyenes are represented by polyene graphs. A (free) polyene graphis defined as one vertex alone or a tree that can be embedded in a hexagonal lattice.Kirby! has recently enumerated the geometrical isomers of these systems by com-puter programming. A complete mathematical solution for these numbers has beenachieved (see Appendix) when it is allowed for overlapping edges and/or vertices onthe hexagonal lattice. Such geometrically nonplanar systems were excluded inKirby's! enumerations.The main topic of the present work are the systems of polyene subgraphs at-tached to one hexagon. These systems represent important conjugated hydrocarbons,of which the smallest systems of the category in question are displayed in Figure 1.POLYENE SUBGRAPHS ATTACHEDTO ONE HEXAGONDefinitions and Introductory RemarksThe Ca+6Ha+6 isomers of the title systems were enumerated by means of gen-erating functions.š " Symbol a is used to designate the number of edges in the ap-pendages. For the sake of clarity, it is repeated that the overlapping edges and/orvertices are allowed, both between the appendages and with the hexagon. The fa-mous paper by Harary and Read,2 who enumerated catafusenes" (a class of poly-hexes) was found to be especially instructive. Also, amore recent work by Zhang etal.' is highly relevant to the present work.64 B. N. CYVIN ET AL.010-lcYOQ-0-COIO-->OF 0-< Q-; eh(YCh-()-lQ--d-~CO=> OJvOF" CYo-<0-\ OJQ-ro.O; d-< o->Q-ro.o,b-< --OJ<:: -0-<~t>(r~gQrCY~J~~cf-<Y-chrO-~-d--6~~-p-~Figure 1. The smallest ea + 6Ra + 6 (O:o; a :o; 4) chemical graphs representing polyenes attachedto benzene. Symmetry types are indicated by: hexagon D6h; triangle D3h; rhomb D2h; dot C2h;vertical arrowhead C2v(b); horizontal arrowhead C2v<a).Auxiliary FunctionsThe basic function Uo(x) is the generating function which counts the Ua rootedunsymmetrical polyene graphs. These numbers are given by the recurrence relationa-1Ua+1 = 2Ua +L UiUa_i (a>l)i=l(1)with the initial conditions Ul = 1, U2 = 2. The relation (1) was derived in the same wayas the corresponding relation for catafusenes.Y By definition, set Uo = 1 (for a = O),cor-responding to the empty graph. The generating function for Ua was deduced as:[SOMERS OF POLYENES ATTACHED TO BENZENE 65co 1 [ 1]Uo(x) = L: U;Xa = "2 x-1 1 - (1 - 4x)2 = 1 + x + 2x2 + 5x3 + 14x4 + 42x5 + ...a=O(2)A similar function, Vo(x), counts the rooted mirror-symmetrical polyene graphs sup-plemented by the empty graph (Vo = 1). It was found:Functions Uo(x) and Vo(x) are analogous to those for catafusene appendages, whichwere designated by the same symbols.Additional auxiliary functions of the form Uo (X~l) vg (XV) are of interest. Firstly,some powers of UO(X) and Vo(x) are needed; then, p = ° or a = 0, respectively. Spe-cifically,V6(x) = x-1 [Uo(x) - 1] (4)ug(x) = x-2[(1 - x) Uo(x) - 1] (5)(6)and(7)Secondly, three useful mixed functions were worked out as:u, (x2) V(x) = x-1 [(1+ x) Uo (X2) - 1]u; (x2) ~(x) = x-2 (1+ 2x) [U (x2) - 1]v6 (X2) ~ (x) = x-4 (1 + 2x) [(1- x2) Uo (X2) - 1] .(8)(9)(10)Preliminary Symmetry ConsiderationsThe systems of polyene subgraphs attached to one hexagon are distributedamong eight symmetry groups, which are specified in the followingtogether with thesymbols used for the corresponding generating functions. The nonplanarity for geo-metrically nonplanar systems is not taken into acount.Regular hexagonal (D6,,) :H(x)Nonregular hexagonal (C6,,) :x (x)Regular trigonal (D3h) : T (x) = Tb (x) + Ta (x)Nonregular trigonal (C3h) : R (x)Dihedral (D2,,) : D (x)66 B. N. CYVlN ET AL.Centrosymmetrical (C2h) : C (x)Mirror-symmetrical (C2J:M(x) = Mb(x) + Ma(x)Unsymmetrical (Cs) :A(x)Functions Mb(x) and Ma(x) pertain to the two types of mirror-symmetrical sys-tems, uiz. C2u(b) and C2uCa),respectively. In a C2u(b) system, the unique twofold sym-metry axis (C2) bisects perpendicularly a central edge, while in C2u(a) the C2 axisgoes through a central vertex. Also, two types, uiz. D3h(b) and D3h(a) are distin-guished among the regular trigonal systems; the se types correspond to Tb(x) andTa(x), respectively. In a D3h(b) system, the three C2axes bisect edges of the centralhexagon, while in D3h(a) the C2axes pass through the hexagon vertices.The generating function V3(x) counts the systems belonging to different symme-try groups a certain number of times according to the relationV3(x) =H(x) + 2X(x) + 2T(x) + 4R(x) + 3D(x) + 6C(x) + 6M(x) + 12A(x) . (11)Hence, for the total number of isomers, the generating function I(x) readsI(x) = H(x) + X(x) + T(x) + R(x) + D(x) + C(x) + M(x) + A(x) =(12)= 112[~(x) + 11H(x) + 10X(x) + 10T(x) + 8R(x) + 9D(x) + 6C(x) + 6M(x)]where A(x) is eliminated from the last expression.Numbers of Symmetrical IsomersIn order to deduce the total numbers of isomers, as given by Itx), it is necessaryto enumerate the isomers for all the pertinent symmetry groups except Cs; ef Eq. (12).In general, certain functions of the type UO(xfl) vg(XV) are employed. Such a functionpertains to a u-tuple unsymmetrical polyene subgraphs and p v-tuple mirror- sym-metrical polyene subgraphs as appendages. The total number of appendages, whichare of interest in the case at hand, isall + pv = 6 (J3)but the empty graph is counted among the Il or v appendages by virtue of the defi-nition Uo = Vo = 1 (see above). The application of the composite generating functionsUO(Xfl) V1l(xV) will be explained by numerous examples in the following. SymmetryD6h is realized by one sextet of identical mirror-symmetrical polyene subgraphs, i.e.p = 1, v = 6. Renee,(14)ISOMERS OF POLYENES A'ITACHED TO BENZENE 67where Eq. (3) has been employed. The hexagonal systems (D6h and C6h) are realizedby sextets of identical unsymmetrical polyene subgraphs (u = 1, ~ = 6). However,function Uo(x6) counts each C6h system twice and each D6h system once. Moreover,there is a one-to-one correspondence between the C6h and D3h(b) systems. Renee,(15)The systems ofthe type D3h(a) are realized by two triplets ofmirror-symmetricalpolyene subgraphs, i.e. J3 = 2, V = 3, but the appropriate function, viz. \1(x3), countsthe D3h(a) systems twice and includes the D6h systems once. Renee,where Eqs. (7) and (14) have been employed. In order to find the numbers of C3hsystems, one should invoke two triplets of unsymmetrical polyene subgraphs (c = 2,~ = 3); in this case, V5(x3) counts each C3h system four times, together with the sys-tems of higher symmetries according to the scheme below.(17)Symmetry D2h is realized by one quartet of unsymmetrical polyene subgraphscombined with a pair of mirror- symmetrical polyene subgraphs. The pertinent sys-tems are counted precisely by UO(X4) Vo(x2), but together with the D6h systems.Renee,where Eq. (8) has been employed, as well as Eq, (14) again.In order to enumerate the C2h systems, consider three pairs of unsymmetricalpolyene subgraphs, giving rise to function V5(x2). It counts systems of several sym-metries various numbers of times according to:(19)68 B. N. CYVIN ET AL.Funetion Vll(x2) eounts also the C2u(b) systems; it has been found:(20)Finally, the C2u(a) systems should be eonsidered. They are realized by two pairsof unsymmetrieal polyene subgraphs and two single mirror-symmetrieal polyenesubgraphs. Renee, function V6(X2) V6(x) from Eq. (10) is to be used. It was found,similarly to Eq. (20),(21)It is evident that the C2u(b) systems, like those of C2h and D2h, oeeur only whena is even-numbered, while the C2uCa)systems oeeur for all a > O. However, for aneven-numbered a, one has the same number of C2u(a) and C2u(b) systems.Total Number of IsomersThe expressions from Eqs. (14) - (21) were inserted into (12). Then, the followingequation emerged for the total number of isomers:whieh is eompatible with Example 4 ofZhang et al. 7 Afurther elaboration of Eq. (22),inserting the expressions in terms of UO(X~I) for every term, yields:(23)Ultimately, we give funetion J(x) in its explieit form, as derived from Eq. (23) bymeans of (2).J(x) = ~ x-6 [12(1 - X2 - 2x3) - (1 - 4x + 3x2)(1 - 4x)1!2 - (7 + 6x)(1 - x2)(1 _ 4x2)1I224(24)The total numbers of isomers from the above analysis are given in Table I to a = 12.The distribution into symmetry groups is included.ISOMERS OF POLYENES ATI'ACHED TO BENZENE 69TABLE INumbers of Ca + 6Ra + 6 isomers representing polyenes attached to benzenea D6h C6h D3h C3h D2h C2h C2u Cs TotalO 1 O O O O O O O 11 O O O O O O 1 O 12 O O O O 1 O 2 1 43 O O 1 O O O 2 8 114 O O O O 1 1 8 31 415 O O O O O O 9 132 1416 1 O O 1 1 4 26 500 5337 O O O O O O 28 1924 19528 O O O O 2 14 88 7216 73209 O O 1 3 O O 89 27194 2728710 O O O O 5 47 292 101978 10232211 O O O O O O 297 383319 38361612 O 1 2 9 5 164 994 1440970 1442145STYRENOIDSThe algebraic solution (24) and the corresponding numbers (Table 1) allow forsystems with overlapping vertices. We shall refer to the pertinent systems withoutoverlapping vertices as geometricalIy planar. It is also reasonable to use the termstyrenoids for the geometricalIy planar systems consisting of polyene subgraphs at-tached to a hexagon. The benzene graph is reckoned among the styrenoids as a triv-ial system of this kind. Styrene (C8R8) represents a nontrivial styrenoid as a proto-type of great interest in organic chemistry. It has been subjected to many studiesduring a long period of time, and a modern structural investigation of this moleculehas appeared very recently.?TABLE IINumbers of Ca + 6Ra + 6 geometrically plan ar isomers representingpolyenes attached to benzene (styrenoids)a D6h D3h C3h D2h C2I, C2u Cs TotalO 1 O O O O O O 11 O O O O O 1 O 12 O O O 1 O 2 1 43 O 1 O O O 2 8 114 O O O 1 1 8 31 415 O O O O O 9 129 1386 1 O 1 1 4 25 470 5027 O O O O O 27 1720 17478 O O O 2 14 80 6069 61659 O 1 3 O O 77 21339 2142070 B. N. CYVIN ET AL.A computer program was implemented in order to generate and enumerate thestyrenoids. The results that were obtained to a = 9, are entered in Table II. The dis-tribution into symmetry groups is included therein; it was determined by a specialalgorithm in the same way as in the case of benzenoids.PGEOMETRICALLYNONPLANAR SYSTEMSThe numbers in Table III were obtained by subtracting the numbers of (geomet-rically planar) styrenoids in Table II from the total s of Table I. Thus, the numbersin Table III count the pertinent geometrically nonplanar systems, uiz. those whichcontain at least one pair of overlapping vertices when drawn on the background ofthe hexagonal lattice.TABLE IIINumbers of Ca + 6Ha + 6 geometrically nonplanar iso-mers representing polyenes attached to benzene56789C2v Cs TotalO 3 31 30 311 204 2058 1147 115512 5855 5867ae"H" 8 8 <388-S-S88888<3SSS-BgS-139S--8p88--8B88--8BB8-Figure 2. The smallest geometrically nonplan ar systems of polyene subgraphs attached to onehexagon.ISOMERS OF POLYENES A'ITACHED TO BENZENE 71Several of the numbers in Table III can be checked by systematic generationson the pen-and-paper level. Firstly, one finds the 3 and 31 systems for a = 5 (CuRu)and a = 6 (C12R12) respectively, as shown in Figure 2. Secondly, the generation ofC2u systems was continued through a = 9, with the results displayed in Figure 3.The smallest C2h systems of the category under consideration occur for a = 10, asshown in Figure 4. This figure includes the smallest of such systems belonging tothe D2h symmetry; it has a = 12. The a value must be increased further in order toproduce the geometrically nonplanar systems with higher symmetries (trigonal andhexagonal), but it is not difficult to draw the smallest of such systems. We omit thedetails.0-00-0 0-0 o--q0Q-cM-QJc88=>8JEP8=>Figure 3. The smallest C2u geometrically nonplanar systems of polyene subgraphs attached toone hexagon.§Figure 4. The smallest C2h and D2h geometrically nonplanar system of polyene subgraphs at-tached to one hexagon.72 B. N. CYVIN ET AL.VALENCE STRUCTURES OF STYRENOIDSThe existence or non-existence of valence structures for the styrenoids is ofprime interest in chemistry. Therefore, the generated styrenoids (ef Table II) wereclassified according to the valence structures. There are three possibilities specifiedin the following.K = o: no valence structure, corresponding to a radical;K = 1: one valence structure with all double and single bonds fixed;K = 2: two valence structures, which arise from an aromatic sextet for thebenzene ring.These three cases were recognized by an extra algorithm on the basis of the de-terminant of the adjacency matrix, which is ](2; hence, it may assume the value ofO, 1 or 4.All styrenoids with odd-carbon formulas, uiz. CnHn where n is odd, have obvi-ously K = O. A styrenoid with an even-carbon formula (CnHn where n is even), onthe other hand, may have either K = O, 1 or 2. The CsHs systems (a = 2) furnishelucidating examples, which are displayed in Figure 5. The results of the classifiedenumeration are shown in Table IV. .K=lFigure 5. The four CsHsstyrenoid isomers, which represent: styrene with K = 2; two moleculeswith K = 1; one radical with K = O.APPENDIXThe Jm numbers of isomers of free polyene graphs with m edges (including thegeometrically nonplanar systems) are counted by a generating function as:J(x) =L Jmxm = 1 +x +X2 + 3x3 + 4x4 + 12x5 + 27x6 +»i v O(25)The following expression for function (25) was deduced, following basically themethods of Harary and Read,2 which imply the Redfield-P6lya theorem? and themethod of Otter+' for passing from rooted to unrooted graphs.Areport on the details of the derivation of Eq. (26) is under preparation.ISOMERS OF POLYENES AITACHED TO BENZENE 73TABLE IVNumbers of styrenoid isomers classified according to valence structuresa Formula K D6h D3h C3h D2h C2h C2v Cs TotalO C6H6 2 1 O O O O O O 11 C7H7 O O O O O O 1 O 12 CsHs O O O O O O 1 O 11 O O O 1 O 1 O 22 O O O O O O 1 13 CgHg O O 1 O O O 2 8 114 ClOHlO O O O O 1 O 2 11 141 O O O O O 1 12 132 O O O O 1 5 8 145 Cl1Hl1 O O O O O O 9 129 1386 C12H12 O O O O 1 O 10 211 2221 1 O O O 4 15 146 1662 O O 1 O O O 113 1147 C13H13 O O O O O O 27 1720 17478 C14H14 O O O O O 7 39 3220 32661 O O O O O 9 1731 17402 O O O 2 7 32 1118 11599 Cl5H15 O O 1 3 O O 77 21339 21420REFERENCE SI. E. C. Kirby, J. Math. Chem. 11 (1992) 187.2. F. Harary and R. C. Read, Proe. Edinburgh Math. Soc., Ser. II 1 (1970) I.3. F. Harary, E. M. Palmer, and R. C. Read, Discrete Math. 11 (1975) 371.4. G. P6lya and R. C. Read, Combinatorial Enumeration of Groups, Graphs and Chemical Com-pounds, Springer-Verlag, New York, 1987.5. B. N. Cyvin,J. Brunvoll, R. S. Chen, and S. J. Cyvin, Math. Chem. (Midheim lRuhr) 29 (1993)13l.6. A. T. Balaban, Tetrahedron 25 (1969) 2949.7. F. J. Zhang, X. F. Guo, S. J. Cyvin, and B. N. Cyvin, Chem. Phys. Lett. 190 (1992) 104.8. S. J. Cyvin, J. Brunvoll, and B. N. Cyvin, J. Math. Chem. 9 (1992) 19.9. R. Hargitai, P. G. Szalay, G. Pongor, and G. Fogarasi, J. Mol. Struct. (Theochem) 306 (1994) 293.10. J. Brunvoll, B. N. Cyvin, and S. J. Cyvin, J. Chem. Inf Comput. Sci. 27 (1987) 171.Il. R. Otter, Ann. Math. 49 (1948) 583.SAŽETAKIzomeri poliena vezanih na benz enB. N. Cyvin, E. Brendsdal, J. Brunvoll, i S. J. CyvinPolienski graf jest stablo koje se dade smjestiti na šesterokutnu mrežu. Razmatraju sepolienski grafovi vezani na jedan šesterokut, pri čemu se dopušta prekrivanje grana i/ili čvo-rova (geometrijski neplanarnih sustava). Nađeno je potpuno matematičko rješenje (u oblikugenerirajućih funkcija) za broj izomera takovih grafova. Za pripadne, geometrijski planarnesustave - sterinoide - broj izomera nađen je računalom. U dodatku je dana generirajuća fun-kcija za broj izomera slobodnih polienskih grafova.

Isomers of Polyenes Attached to Benzene

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