graphs in chem

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334 J . Chem. InJ Compur . Sci. 1985, 25, 334-343Applications of Graph Theory in Chemistry

ALEXANDRU T. BALABANDepartment of O rganic C hemistry, Polytefhnic Institute. 76206 Bucharest, Roumania

Received March 12. 1985Graph theoretical (GT) applications n chemistry underwent a drama tic revival lately. Con-stitutional (molecular)graphs have points (vertices) representing atoms and lines (edges) sym-bolizing ma le n t bonds. This review deals with definition. enumeration. and systematic codingor nomenclature of constitutional or steric isomers, valence isomers (especially of annulenes).and condensed polycyclic aro matic hydrocarbons. A few key applications o f graph theory intheoretical chemistry are pointed out. The complete set of al l poasible monocyclic aromatic andheteroaromatic compounds may be explored by a mmbination of Pauli's principle, P6lya's theorem.an d electronegativities. Topological indica and some of their applications are reviewed. Reactiongraphs and synthon graphs differ from constitutional graphs i n their meaning o f vertices andedges and find other kinds of chemical applications. This paper ends with a review of the useof GT applications for chemical nomenclature (nodal nomenclature and related areas), coding.and information processing/storage/retrievalINTRODUCTION

All structural formulas of covalently bonded compounds aregraphs: they are therefore called molecular graphs or, better.constitutional graphs. From the chemical compounds de-scribed and indexed so far, more than 90% are organic orcontain orga nic ligands in whose constitutional formulas thelines (edges of the graph) symbolize covalent two-electronbonds and the points (vertices of the gra ph) symbolize atomsor, more exactly. atomic cores excluding the valence electrons.Constitutional graphs represent only one type of graphs thatare of interest to chemists. Othe r kinds of graphs (synthongraphs. reaction graphs, etc.) will be mentioned later. Thisreview will try to highlight applications of such graphs inchemistry: grap h theory provides the basis for definition,enumeration, systematization, codification, nomenclature,correlation, and computer programming.'-'Chemistry isprivileged to be, both potentially and actually,the best documented branch of science. Th is is mainly dueto the facts that m ost of the chemical informa tion is associatedwith structural formulas and that structural formulas may besystematically and uniquely indexed and retrieved. In othe rbranches of science. one needs to use words, but in mostbranches of chemistry, chemists may 'talk" with one ano therby means of formulas only. irrespective of their mother tongue.Of course, one does translate chemical structures into wordsby means by nomenclature rules, but the time has come whencomputers can manipulate formulas directly; moreover,chemistry does possess interdisciplinary are as bordering withphysics or biology where words are indispensable; one maythink about any property or application associated with achemical structure, and then it is clear tha t af ter locating acompound in a datab ase one needs words. On e should note,however, that in chemical information words usually comeafterward, (the search field being considerably narrowed),whereas in all other fields they come first.The importance of graph theory (GT) for chemistry stemsmainly from the existence of the phenomenon of isomerism,which is rationalized by chemical structu re theory. This theoryaccounts for all constitutional isomers by using purelygraph-theoretical methods, which earlier chemists viewed as'tricks with points an d lines (valencies)". It is difficult now-aday s to realize the awe and wonder this theory caused 125years ago; the mira cle was repeated twice: first when thestereochemistry of carbon compounds was elucidated by LeBel and Van't Ho fP and then when Werne r9 extended theseideas to metal complexes.Defining and finding the constitutional isomers (whichcorrespond to the sa me given molecular formula) a re purely

0095-2338/85/ 1625-0334101.50/0

Alexandru T. Balaban. born in 1931. is Professor of OrganicChemistry at the Polytechnic Institute in Bucharest. He graduatedin 1953 and obtained the Ph.D. degree in 1959 with a thesis onAICI,-catalyzed reactions (supervisor, C. D. Nenitzescu). He isa corresponding member of the Academy of S. R. Roumania since1963 and a fellow of the New York Academy of Sciences since1969. He has published more than 350 papers or book chaptersand has edited books on labeled compounds (1969,1970) and thebook "Chemical Applications of Graph Theory" (AcademicPrrss:London, 1976). He is author of six other books, primarily onorganic chemistry topics.graph-theoretical problems. Chemists, however, have usedunconsciously graph-theoretical concepts for more than 100years, just as M oliere's "bourgeois gentilhomm e" was speakingprose without realizing it.Th e problem of isomerism is the real crux of the docu-mentation/retrieval problem. Although molecular formulascan be ordered for indexing purposes according to simple NIS(alphanumerically, except for carbon and hydrogen atomswhen present). there is no simple and ev ident means of orderingconstitutional graphs with the same numbers and types ofvertices, which represent isomers. It is here that more so-phisticated techniques of GT may help.The essence of chemistry is the combinatorics of atomsaccording to definite rules: hence, the adequate mathematicaltools ar e graph theory and combinatorics, which ar e closelyrelated branches of mathematics. Until recently, most theo-retical chemists viewed mathematics as a tool for professing

0 1985 American Chemical Society


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APPLICATIONSF GRAPH HEORYN CHEMISTRY J . Chem. Inf . Comput. Sci., Vol. 25, No. 3, 1985 335VALENCE ISOMERISM

A s ubcla ss of constitu tional isomers are the valence isomers,defined classically* as isomers differing by shifting th e pos-itions of simple or double bonds. A clearer usesG T concepts: if two hydrogen-depleted graphs have the sam epartition of vertex degrees, they co rrespo nd to valence-isomericmolecules; for vertex -labe led graph s (i.e., molecules conta iningheteroatoms such as 0,N , etc.), vertex degrees associated witheach label (atom type) must be conserved. Examples for thesmallest alkane valence isomers exist among th e five isomerichexanes: one Me2 (CH 2)4, ne Me4(CH )2,one Me4(CH 2)C,and two valence-isomeric Me3(CH2)2(CH)(10 an d 11).L - p v g ? $ v1L 150 11 12 13

Counterexamples for heteroatom-containing molecules aredimethyl ether (5) nd ethanol (6),which are isomeric butnot valence isomeric because vertex degrees are not conservedfor each atom type separately. There exist four valence isomersof furan, thiophene, or pyrrole, (CH )4X ,(12-15),X =0, ,or NH;21-23n these cases when multiple bonding may exist,one should replace the ter m graphs by multigraphs in orderto be more precise.It should be noted from examples 1-15 that the sum of allvertex degrees is identical for isomers; valence isomers are thoseisomers where the summ ands a re correspondingly equal foreach type of atom.Th e most inter esting case of valence isomerism is providedby annulenes, (CH),, whose hydrogen-depleted graphs areregular graphs of degree 3 (cubic graphs). A simple graph-theoretical theorem states that the number of vertices inregular grap hs of odd degree must be even; therefore, cubicgraphs , (CH),, must have an even n. From the six possiblecubic graphs with six vertices, 16-21, he last one, 21, s

(crunching) numerical data, but the present trend towardnonnumerical method s is noticeable, mainly due to the impactof GT.CONSTITUTIONAL AND STERIC ISOMERISM

In graph theory the number of lines meeting at a vertex,i.e., incident to th at vertex, is called the vertex degree; graphswhose vertex degrees are all equal ar e called regular graphs.Th e mathema tician Cayley proposed 128 years agolo analgorithm for calculating the num bers of constitutional isomersfor alkanes, CnH2,+2, nd alkyl derivatives, e.g., C,H2,+1Cl.In graph-theoretical term inology, the latter are called rootedtrees with the s ame vertex degree restrictions. Of course, then vertices of degree 4 correspond to carbon and the 2n +2vertices of de gree 1 to hydrogen atoms. Alternatively, one mayignore all hydrogens and look for the equivalent set of treeswith n vertices of degrees 1-4; edges now correspond exclu-sively to C-C bonds, and together with the vertices they formth e hydrogen-depleted graph, or skeleton graph, of the mol-ecule.In 1931-1933, the chemists Henze and Blair corrected afew mistakes in Cayleys enumerations and thus perfected analgorithm, which nowadays is easily implemented by com-puter progra ms for various types of cons titutional isomers ofalkyl derivatives.12 A mathematically elegant approach forthe sam e problem was presented by the mathematician P dlyain 1936-1937 when he formulated the important theorem,which since then bears his name.13,14It may sound a trivial difference, but there is an importantdistinction between todays formulation (i) that there existtwo isomers each of C 4H 10 1 an d 2), C3H7C1 3 an d 4), an d

Graphs- >H,CICH,),CH, IH,CI,CH

1 2Rooted Graphs,

v v v v(H,C) ,CHCI ClCH,CH$H, C O C C C 0

H 3 H3 H 3 H 23 L 5 6

C 2 H 6 0(5 nd 6) ecause G T demonstrate^'^ that there existexactly two trees on four vertices and two rooted trees on threevertices, and (ii) the old textbook approach saying if one triesto write constitutio nal formulas, one finds only two isomers.Th e difference is the sam e as between the correct formulation(i) the 1st and 2nd principles of thermodynamics assert theimpossibility of a perp etuum mobile of 1st and 2nd kind and(ii) until now, it has not yet been possible to obtain a per-petuum mobile of 1st or 2nd kind.Problems of stereoisomerism are also elegantly solved bygroup-theoretical and graph-theoretical techniques (althoughgraph theory, like topology, ignores geometrical distances andangles) by app lying Pdlyas13 or Otters theorernsl5 to objectswith the appropriate (geometrical) symmetry groups. On eshould, therefore, similarly say GT demonstrated that (i)there exist two smallest chiral heptanes 7 an d 8 where theJ 6Y -XH7 8 9

chirotopic/stereogenic16 arbon atom s are shown with a hy-drogen atom, and that (ii) the smallest meso-alkane is anisomer 9 of octane.

16 17 18 191 1 9

20 21A 2 18nonplanar, i.e., cannot be written on a plane without crossinglines. No such molecule can exist.In addition to those six (C H), graphs, there are 21 1 otherC6H6 graph^^^,^^ that are isomeric (not valence isomeric!) tobenzene. They were all depic ted in refer ence 23 and later wereincluded in advertisements of the Jeol A few examplesof such isomers are fulvene (22) nd the three hexadiynes(23-25).

22 23 2.4 25Two graphs are isomorphic if the re exists a vertex labelingthat preserves adjacency; they can be viewed as differentgeometrical representations of the sam e abstrac t graph definedas a set of elements (vertices) {v i) , E 1, 2 , ..., n , and a setof elements (edges) that are unordered duplets from the formerset {upj), #j E 1, 2, ...,n. Thus, 21A and 21B are isomorphicgraph s. If the restriction i # j is removed, Le., if one allowsa vertex to be connected to itself, one obtains loop graphs; ifboth loops and m ultiple edges are allowed, one obtains general


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336 J . Chem. In$ Comput. Sci., Vol. 25 , No. 3 , 198.5graphs having edges u p , (loops, which are counted twice inthe de gree of vertex i) and m ultiple edges up,, vp,, ... (twiceor thrice).There a re two fundamental nonplanar graphs, 21 an d 26,depicted each in two igomorphic representations (A and B ).

BALABAN

2 6 A 26 8 27Kuratowsky demonstrated26 hat a necessary and sufficientcondition for a graph to be nonplanar is to contain 21 or 26as a subgraph. If one removes the restriction tha t a graph edgesymbolizes one covalent bond and allows an edge(s) to sym-bolizing a string(s) of several covalent bonds, then homeo-morphic graphs ar e obtained. During the last years, severalmolecules homeomorphic with nonplanar graphs have been~ b t a i n e d . ~ Coming back to valence isomers of annulenes, it was shownby a con stru ctiv e a l g ~ r i t h m ~ - ~ ~hat there exist two planarcubic multigraphs for (CH),, five for (CH),, 19 for (CH)g,7 1 for (CH),,, and 35 7 for (CH),,. A computer programTising an elaborate formula for general cubic multigraphs wasdeveloped for checking th e above numbe rs;28 one must cal-culate separately by recurrence algorithms and subtract thedisconnected graphs, the loop graphs, and the nonplanar graphsfor arriving at the above numbers. In the formulas that werepublished for the valence isomers of ann~lenes,-~~enzo-, n n n ~ l e n e s , ~ * ~ ~nd homo-*S3@r he teroannulenes,3 no detailedconsideration was given to stereoisomerism so that actuallythe numbers of possibilities ar e higher. Exciting experimentaldiscoveries in this field occurred since 1961 when Nenitzescu,4vram, and co-workers3* ynthesized the first valence isomerof an annulene. a hydrocarbon (CH),, named by Doeringq3:i s Nenitzescus hydrocarbon (27). Tetra-tert-butyltetra-hedrane and its cyclobutadienic valence isomer were preparedhy Maie r et a l.34 nd shown to be sta ble, unlike the less sub-stituted deriva tives. All five benzene valence isomers, 16-20,are known (bicyclopropenyl only as substituted derivatives);Van Tamelen, Kirk, and Pappas prepared Dewar benzenederivatives (a misnomer .that has gained large cir~ulation);~Wilzbach and K a ~ l a n , ~ ~hen Katz,j7 prepared benzvalene;wnzprism ane and benzvalene isomerize explosively to benzene .Bullvalene, (CH),,, predicted by Doering and Rothq8 ndLynthes ized by S~ h r O d e r ,~ ~nd semibullvalene, (CH)s,40 n-dergo rapid automerizations; cuba ne, (CH),, pentaprismane,(CHJ,,. both synthesized by E, ton,4 and dodecahedrane,:CH),,, are spectacular cage-type molecules. The latter wassynthesized stepwise by P aqu ette et aL4, and via isomerizationDY Schleyer et al.43 Many of these valence isomers presentconsiderable interest because they illustrate the brilliantpredictions based on the Woodward-Hoffmann rules.4J

An open problem is the naming of these compounds. Asillustrated by the rapidly increasing numbers of planar cubicinultigraphs (chemically possible valence isomers of [n]-s nnulenes) with increasing n, it is not reasonable to coin trivialnames (cuneane, hypostrophene, snoutene, etc.); on the otherhand, the systematic I UP AC names using the Baeyer systemare too cumbersome. I t was pr op o~ ed ~ -~ *o use a string ofSgures fo r characterizing such [nlannulen e valence isomers;these figures indicate, in order, the numbers of vertices, doublebonds, and three- and four-membered rings and a serialnumber according to certain conventions. Th e only drawbacko f thi s sys tem, which was adopted by F a r n ~ m , ~ ~ones andand other che mists interested in valence isomers ofdnnulenes, is that one needs the table of possible valencei c m x r s for retrieving the structure from the notation.

The valence isomers of furan 12-15 can be derived fromgeneral cubic graphs by obtaining all such graphs with oneloop22and by assimilating the loop and the vertex to whichit is bonded with the heteroatom X . On the other hand,structures of valence isomers of pyridine and other azines maybe obtained3 by applying Pblyas theorem to valence isomersof benzene.The most ambitious GT-based computer program for isomergeneration develop. d so far was initiated by Lederberg,Djerassi, Sutherl and, Feigenbaum, Duffield, and associate^:^^DENDRAL and its heuristic successor programs are able todetermine the chemically reasonable structures for a givenmolecular formula. In conjunction with mass spectrometricand N M R da ta, this program could enable a completely au-tomatic struc ture determination for any substance with mo -lecular mass lower than about 1000. The continuing searchfor extraterrestrial life forms needs such determinations.

CONDENSED POLYCYCLIC AROMATICHYDROCARBONS (PAHS)The classical definitions of ca ta- a nd peri-condensed PAHsinvolve the absence a nd presence, respectively, of carbon atomscommon to three rings.47 Though this definition clearly in-dicates naphthalene (28), phenanthrene (29), and anthracene

28 29 30

31 32 33(30) to be cata-condensed and pyrene (31) or perylene (32)t o be peri-condensed, it is ambiguous for the rec ently pre-pared48kekulene (33). A graph-theoretical definition49s basedon the dualist graph (th e graph haqing as vertices the centersof benzenoid rings, with two vertices being connected if thecorresponding benzenoid rings a re condensed): cata-condensedsystems have acyclic dualist graphs, peri-condensed systemshave dualist graphs possessing only three-membered rings, andcorona-condensed systems have dualist graphs possessing largerrings (in this series the last criterion prevails, i.e., peri-con-densed systems may have cata-condensed portions, whilecorona-condensed systems may have both cata- and peri-condensed portions of the molecule).With this definition, all cata-condensed systems with thesame number p of benzenoid rings are isomeric. G T methodsprovided enumerations and structures of all cata-condensedsystems with p


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APPLICATIONSF GRAPH HEORYN CHEMISTRY J . Chem. Inf. Comput. Sci., Vol. 25, No. 3, 1985 337Topological correlations exist also with reactivity da ta, e.g.,with reaction rates for the Diels-Alder cycloaddition of maleicanhydride to cata- and perifusenes; the correlations considertwo parameters: the number of benzenoid rings in the longestcata-condensed linear portion and a shape parameter (an in-teger number expressing the nature of the annelation at theends of this largest linear portion). The agre ement with ex-periment for more tha n 100 cata- and perifusenes is better6than in any other correlation using more sophisticated theo-retical calculations.66The topology is also important for the carcinogenic activityof PAH s, where the presence of bay regions is responsible forthe conversion of an adjacent benzenoid ring into a diol ep-oxide, probably the ultimate carcinogen. A com puter programgenerates all PAHs, recognizes the presence and number ofbay regions, and prints the result and stru cture of the PA Htogether with its three-digit code.67By ignoring the difference between digits 1 and 2 in 3DC ,Le., between annelation kinked toward right or left, and byreplacing both 1 and 2 by 1 (Le., by kinked annela tion of anykind), one obtains a two-digit notation that no longer codesthe complete structure but only its topology: this is called theL transform of the 3D C into a 2DC since the resulting codeconsisting of 0 and 1 can be still interpreted as a binarynumber.68 Alternatively, Gutmad9 ntroduced an analogous

description of the topology in terms of the LA sequence (L=linear annelation, A =angular an nelation); the sequenceof L /A letters is completely equivalent to the seque nce of 0/1letters/digits. Catafusenes having different 3DC but identical2DC were termed70 soarithmic because they give rise to th esam e number of Kekul& tructures. Moreover, many of theirspectral or chemical properties are quasi-identical, e.g., elec-tronic and photoelectronic spectra, rate s of Diels-Alder cy-cloaddition with maleic anhydride , etc. Examples of isoar-ithmic catafusenes are tetrahelicene and chrysene, pentahel-icene (39), picene (40), and hydrocarbon 41, and the pair ofdibenzanthracenes 42 an d 43.

On the basis of specifying in digital form the th ree possibleorientations (34) of carbon-carbon bonds in the graphite

5 L Q 3 \a1212123121312323 1212123131212323

34 35 36l a t t i ~ e , ~ne may obtain a coding of planar annulene ge-ometries (e.g., 35 and 36) that is free from ambiguities existingin other codes.56 On e selects a vertex and a sense of rotationso that the resulting string of digits forms the minimal numberfor the given geometry. On t he sam e basis, TrinajstiC, Knop,and co-workerss7 developed algorithms and computer programsfor the construction and enumeration of all polyhexes. Diass8has studied th e homologous series of polyhexes as functionsof their molecular formulas and numbers of benzenoid rings.Two different approac hes have been proposed for the codingof perifusenes: one may specifys9 he presence of benzenoidrings (vertices of dualis t graph) in concentric shells (37) around

3 7the topological center of th e polyhex; the ce nter vertex is notspecified in the notation; the first shell has numbers 1-6wherein 1 lies left from the center on a horizontal line; thesecond shell has numbers 1-12; etc. Shells are delineated byslashes, and the polyhex is oriented so as to minimize thenumbe rs in the first shell according to certain rules, as seenin the tw o examples beside 37.Alternatively,@ one may superimpose th e polyhex on thegrid numbered 38 so as to minimize th e area of th e circum-

38 [1 31-triacene 1361-tetracene 2H-12 7ktetracenescribed parallelogram and to minimize the number of rowsand, at equal number of rows, the number of columns.Th e presence of a vertex of a dualist grap h is read as 1 andthe absence as 0, yielding rows of binary numbers that, whentranslated into decimal numbers, afford a coding of the po-lyhexe60GRAPH THEORY AN D THEORETICAL CHEMISTRY

It is a known fact61 ha t the simple Hiickel MO theory andth e VB theory a re both topologically based. In particular, theH M O eigenvalues (in @ units) ar e identical with the eigen-values of the adjacency matrix of the hydrogen-depletedgrap he6* It is therefore not surprising that much chem icalinformation on cata-condensed systems is encoded in the to-pology and geom etry of their dualist graphs (actually, dualistgraphs a re exceptional graphs because their angles do matter) .Thus, it was found that the resonance energies and redoxpotentials are linear functions of two variables: the num berp of condensed benzenoid rings and the number s of 180angles in the dualist graph or of zeroes in the 3DC.63*64 helatter num ber indicates how m any anthracenic subunits (whichmay occur separately or overlap in longer linear portions) existin the cata-condensed PAH; Le., phenanthrene has s =0,anthracene and benzo[a]anthracene have s =1, tetracene hass =2, etc.

3DC 111 121 112 10 2 1012DC 111 111 111 101 101

39 LO L l L 2 43The similarity of chemical and physical properties forisoarithmic catafusenes demonstrates that the topology encasedin the 2D C or th e LA sequence is sufficient for determiningthe frontier M O s . However, none of the isoarithmic cata-fusenes appear to be similar on using the simple MO methods;on the other hand, the known cospectral (or isospectral)gra phs corresponding to molecules that possess the same ei-genvalues of the Hiickel or adjacency mat rice^^'.^^ have dif-ferent physicochemical behavior, as pointed out by Heil-b r ~ n n e r . ~ ~xamples of cospectral graphs (th e term o riginatesfrom th e equality of the eigenvalues) are 2-phenyl-1,3-buta-diene (44) and 1 Cdivinylbenzene (45); the corresponding

LL L5molecules have different electronic absorption spectra andchemical behavior. These dat a prove the deficiencies of thesimple MO methods and point to th e need for a theoreticalrationalization.Dewar74 rovided a major remedy for one of the deficienciesof the Hiickel MO approach by redefining the standard againstwhich resonance energ ies must be calculated; instead of disjoint


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338 J . Chem. In$ Comput . Sci., Vol. 25 , N o. 3, 1985 BALABANTable I. Total Numbers K , ,kof KekulB Structu res an d R,:kofConjugated 6 Circuits (Perfect Benzenoid Rings) and Ratio betweenConjugated 6 Circuits and Total Number of Benzenoid Rings forHelicenes and Isoarithmic Polvhexes

k1 2 4(28) (29) 3 (37-39) m

K l , k 3 5 8 13R l . k 4 10 20 40R i l / f ( i k+l)K,] 0.667 0.667 0.625 0.615 0.553

Table 11. Numbers RIk of Conjugated t Circuits in (1.k) Hexesk

t 1 2 3 4 56 4 10 20 40 7610 2 4 10 20 4014 2 4 10 2018 2 4 10

ethylene units, one takes connected acyclic graphs with thesame number of a-centers and a-electrons. The Dewar res-onance energy, originally computed according to Poples S C Fparametrization, was subsequently simplified by making H M Ocalculations with new standards and parametrizations. TheTrinajstiE-Gutman topological resonance energy,75 theH es s -Sch aad p a ram e t r i ~a t i o n , ~~erndons structure-reso-nance theory,77 nd the conjugated circuits model developedby RandiE78 and independent ly by Gomes and M a l l i ~ n ~ ~lllead to consistent results. All these newer theoretical ap -proaches have a G T basis and ma ke explicit use of G T: byapplication of the Sa chs theorem, eigenvalues may be readilycalculated by recursive formulas; Herdons revival of va-lence-bond methods based on KekulC struc ture counts benefitsf ro m i nt er es ti ng a l g ~ r i t h m i c ~ ~r algebraic70 procedures forcomputing numbers K of KekulC structures. The conjugatedcircuits model was applieds0 o zigzag (j, k ) catafusenes havingequal numbers j of benzenoid rings in each of their k linearportions (or to all their isoarithm ic isomers): if j = 1, oneobtains, as known, the Fibonacci sequence, K l , k=F k f 2 Fo=F , =1). Interestingly, the ratio between the numbers Rj,kof conjugated six circuits (perfect benzenoid rings, or fullybenzenoid rings) having three conjugated double bonds andthe tota l number of six-membered rings, Kj,k.(jk+ l) , givesa simple expression when j increases indefinitely:

Thus, phenanthrene (29) has K,,*=5 KekulC structures witha total of R,,2= 10 perfect (fully) benzenoid rings. Ot hervalues ar e seen in Table I. Values Kj,k for k >1 may beconsidered to be generalized Fibonacci numbers.It was establisheds0 h at the sequen ce of conjugated t circuitsR\,k in U,k)hexes is the sam e for t =10as for t =6 but shiftedas seen in Table 11;e.g., for picene and its isoarithmic ( j , k )hexes with k =4 and five six-membered rings, the numberof fully aroma tic rings in all 13 KekulC structures (conjugated6 circu its) is 40, the num ber of conjugated 10 circuits is 20,etc. The se da ta allow the easy calculation of resonance en-ergies of such (j,k)catafusenes according to RandiEs para m-etrization modified so as to consider all conjugated circuits,not only the linearly independent ones.78HEURISTIC COMBINATORICS FOR M ONOCYCLICAROMATIC SYSTEMS

In accordance with Paulis principle, sp2-hybridized atomsthat form a planar a romatic ring may have 2, 1, or 0 electronsin the nonhybridized pz orbi tal; therefore, they ar e of exactlythree types: X, Y , or Z , respectively. If the ring is M sized

Table 111. First Row Atoms That M ay Form A romat ic RingsTogether with Aromaticity (Electronegativity) Constant (kH) inParentheses) for the C ase R =H (Otherwise, kH =kR +20up,,)grOUDa

type 13 4 14 5 15 6

k R(28) (50) (100)

I n this paper the periodic group notation is in accord with recentact ions by IUP AC and AC S nomenclature commit tees. A and B no-tation is eliminated because of wide confusion. Gro ups IA and IIAbecome groups 1 and 2 . Th e d-transition elements comprise groups 3through 12, and the p-block elements comprise groups 13 through 18.(Note that the former Roman number designation is preserved in thelast digit of the new numbering: e.g., 111- and 13.)and if only first row atoms are involved (Table 111), Huckels4 N + 2 a-electrons rule holds; therefore, for an aromaticX,Y,Zz ring we have

x + y + z = M2x +y =4 N +2

For given M and N values, one can construct all possiblemonocyclic arom atic system s by solving algebraically and thencombinatorially either directly* or by means of Pblyastheorems2 the above system of two equations with thre e un-knowns x, y , and z (which admits several solutions). For givenx , y , an d z values, we arrive at the typical combinatorialproblems known as the necklace problem in GT, whichconsists of finding how many nonisomorphic necklaces one canform with given numbers x , y , and z of differently coloredbeads. Tab le IV presents all solutions for a a-electron sextetwith no adjacent Z-type atoms. The combinatorial problemwith adjacency restrictions was solved by Lloyd.83The complete set of solutions evidences many unknownsystems, especially with boron atom s. However, some of themare too unstable because of unsuitab le electronegativities orbecause they ca n dissociate into stable molecules (e.g., hexazin e- N2).84A semiempirical measure of electronegativities forthis case was proposed,85 ssigning to each a tom type in TableI11 a con stan t; sum mati on of these c onsta nts over the wholering should give a su m within certain limits; otherwise, the ringis unlikely to be aromatic and stable.Consideration of Table IV leads to a generalized definitionof mesoaromatic systems indicated by a n asterisk (mesoioniccompounds are a particular case): aromatic systems havingodd numbers of Y-type atoms separated by X- and/or Z-typeatoms or chains of such a toms ar e called mesoaro matic sys-tems. The carbonyl carbon atom behaves as a Z-typ e atom.S o far, many five-membered mesoionic systems have beenprepareds6 and a few 1,3-diboretess7 (which ar e more stablethan 1 Zdib oret es), but six-membered systems should also bepossible; several such systems can be seen in Table IV, andby combina tion of this tab le with T able 111, many possibilitiesare generated.

GRAPH TH EORY, TOPOLOGICAL INDICES, ANDQSARFor quantitative structureactivity relationships, which areimportant especially for drug design, one looks for correlations


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APPLICATIONSF GRAPH HEORYN CHEMISTRYbetween physical, chemical, or biochemical properties (pos-sessing a continuous variation and being expressed numeri-cally) and chemical structure s. Only a few chemical feature smay be expressed numerically: electronic effects of substit-uents (Hammett's u constants), steric parameters (Taft's orrelated constants), and hydrophobicity (Hansch's partitioncoefficient). Usually, one has to compar e closely relatedmolecules in order to obtain meaningful correlations.A different approach was initiated by H . W iener in 194788with topological indices (TI's). These TI'S are numbers as-sociated with chemical structures via their hydrogen-depletedgraphs G. For hydrocarbons, the Wiener index W is the sumof the num ber of bonds betwe en all pairs of vertices in G. fone defines the (topological) distance betwee n two vertices ofa graph as the number of bonds along the shortest path be-tween these two vertices, then W is the sum of all distancesin graph G. On e can associate with any graph on n verticesseveral matrices: the adjacency matrix A(G) is a square,symmetrical, n X n matrix with entries ai i =1for adjacent(directly bonded) vertices i an d j and zero otherwise; thedistance matrix D(G) s also a square n X n matrix with entriesdii=0 on the main diagonal and di j=1 for adjacent verticesi and j as in A, but all other entries ar e integers bigger than1and represent the topological distance between vertices i andj . The sums over rows i or columns i fo r A(G) indicate thevertex degrees vi; he sum s over rows i or columns i for D(G)indicate another graph invariant for each vertex (invariantfrom the arbitrary vertex labeling i E (0 , 1, ..., n)) , calleddistance sum s i . It is easy to see that W =Cs. .More than 40 other TI'S were devised till now. 9-92 Fromthem, R andiE's molecular connectivityx,93,94 o s ~ y a ' s ~ ~ndexZ , a nd th e re ce ntly i n t r ~ d u c e d ~ ~ndex J (distance sum con-nectivity) ap pear to be the most useful:

I I!

x = C ( v p p= C p ( G , k )4W-q

S

k=O

J =q C ( d i d j ) - 1 / 2 / ( q n +2)In the above formulas, Edgestands for sum mation over allq edges. By multiplying the vertex degrees or the distance sumof end points for each edge of grap h G on n vertices, this maybe chosen so that no two of them have a common vertex: bydefinition, p(G,O) = 1 an d p(G,l)=q, the number of edges;s s the maximu m number of e dges disconnected to each other.A general feature of all topological indices is their degen-eracy, i.e., the fact tha t two or more differen t graphs have thesame value of a TI. The degeneracy is high for Wand 2, lowerfor x, nd very low for J . For reducing dege neracies, Bonchev,Mekenyan, and TrinajstiE devised information theoreticalin dice^^^*^* a nd prop ose d a ~ u p e r i n d e x ~ ~onsisting of a su mof 10 different TI's.There exist correlations among various TI'S.^^ The sys-tematic exploration of molecular properties that are moreclosely associated with TI'S revealed t ha t the molecular re-fractivity and the molecular volume are such properties;therefore, TI'S may be useful param eters for expressing stericfeatures of molecules or substituents.ImVarious TI's have different correlating abilities withphysical, chem ical, or biological prope rties of molecules. Thu s,W a n d Z account well for thermodynamic properties of al-kanes. Ce ntric indices,91 which were not discussed above,correlate well with oc tane numbers of alkanes but have highdegeneracies.101However, it was with RandiE's index that m ostcorrelations were made so far, especially for biochemicalproperties and dr ug design.94 T he presence of heteroatoms

edges

J . Chem. Inf. Comput. Sci., Vol. 25 , No. 3, 1985 339along with carbon atoms requires special treatment, which isavailable till now only for indices x94 nd J. lo2

SEARCH FOR GR APH INVARIANTSTh e problem of cospectral graphs is connected to the searchfor simple grap h invariants, i.e., mea ns of cha racter izing grap hsup to isomorphism. It had been conjectured that the set ofeigenvalues or the charac teristic polynomial of a graph (ob-tained from the adjacency matrix by inserting x on the maindiagonal and by solving the resulting determin ant) is such aninvariant;lo3however, Balaban and Har ary7 [ provided coun-terexamples of cospectral hydrogen-depleted graphs. Then,it was again asserted tha t if hydrogens are included, a gra phinvariant may resultl@ but Herndon105proved the fallacy ofthis idea. Thus, so far, short of the adjacency matrix orequivalent data, a graph cannot be characterized by simplegraph invariants. For establishing whether two graphs are orare not isomorphic, no satisfactory algorithm exists for whichthe time required should increase less fast than polynomiallyvs. the number of vertices.Since the adjacency matrix A(G) characterizes completelya graph G, andiEIM searched a standar d form for presentingA(G). By permuting rows and columns of A(G) and byreading the rows sequentially, one obtains various binary

numbers; if one selects the smallest number, then one reallyhas a graph irlvariant and a binary notation of the graphstructure. However, there exist two problems: namely, (i)the long search for the minimal binary number that isequivalent to the search for a vertex labeling (however, RandiEgave several rules for shortening this search) and (ii) thepossibility of local minim alo7 ha t prevent one fro m reachin gthe real minimum (it appears, however, that for co nstitutionalgraphs such cases are unlikely and that triple permutationsof rows or columns may provide a way out from local mini-ma).1o8 Each vertex of a graph may be ch ar a~ ter ize d'" ~ya set of integers representing the num bers of paths with variouslengths to the other vertices in the graph (a path is a contin-uous sequence of grap h edges so that all vertices are distinct).An efficient algorithm and computer program exist for ob-taining the path num bers for graph vertices. Another idea dueto RandiC is to combine the topological index x with the pathnumbers in order to obtain a molecular identification number,ID."O Howeve r, for multigrap hs, the ID does present de-generacies. It is conjectured"' that if one adopts a similarapproach based on distance sums and path numbers insteadof vertex degrees and pa th nu mbers, the de generacy woulddisappear. Further a ttempts to devise practically nondegen-erate topological indices are based on hierarchically orderedextend ed connectivities (see below). Finally,"[ one maycombine just two TI's, namely, J summed with an index basedon the nu mbers of second (once removed) neighboring vertices.

REACTION GR APHSConstitutional graphs a re not the only types of graphs tha tar e of interest for chemistry. If, in order to overview thesuccessive intermediates in m ultistep reactions o ne depicts bya point (vertex) each intermediate and by a line (edge) anelementary reaction step (the conversion of one intermediateinto the next), one obtains a reaction graph. The first reactiongraph to be published described' l 2 the intramolecular isom-erization of a pentasubstituted ethyl cation, 46, with five

L6 L7 L8 49


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340 J . Chem. Znf. Comput. Sci. , Vol. 25, N o. 3, 1985

x --Im4 X-2

3 c 1

BALABAN

_______^-3

x -x X-( 2 2 :

(1 4 3:Mesoaromatic compounds are indicated by an asterisk, and Y-typeatoms are not shown explicitly in order to avoid croding the formulas.

different graphs and with the two carbon atoms distinguishable(e.g., by isotopic labeling). If, however, there is no means ofdiscriminating the two carbon atoms, th e graph reduces froma 20-vertex graph (47) to a 10-vertex graph 48 (the Petersengrap h or the five cage), which was also discussed by Du nitzand Prelogl 3 in the context of intramo lecular rearrangem entsfor five-coordinated trigonal bipy ramida l (TB P) species, ig-noring their stereoisomerism. If stereoisomerism is considered,the rearran geme nt of TB P complexes is described by graph48, which was used in the context of phosphoranes, 49, byRamirez and co-w ork er~ ."~ he isomorphism between thepentasubstituted phosphorane graph and t he pentasubstitutedethyl cation graph (all substituents being different) was pointedou t by Mis1 0w.I~ ~ n describing intramolecular rotations,Mislow used reaction graphs. A spectacular success of reactiongraphs was in providing a plausible structure for an inter-mediate product in the isomerization leading to diamantan e,which was discovered and investigated by Schleyer and co-workers.l16 The corresponding reaction graph has >40 00vertices, whereas that of the related isomerization affordingada ma nta ne involves a reaction graph"' consisting of 16vertices and 2897 possible reaction paths. For coping with th eformer graph, a computer program was devised in order toselect at each step the energetically most favored path.Il6 On eshould note in this context that the systematic enumerationof "diamond hy drocarbons" is the analogue in three dimensionsof the two-dimensional catafusene problem and can be solvedby applying the dualist-graph approach."*Further uses for reaction graphs were for intramolecularrearrangements of octahedral complexes or for degenerate

Targetmolecule

Synthon' p raphs2graphsOptimalplanning

c1 3CL 3Figure 1. Synthon graphs (second row) for the same target molecule(first row) dissected differently. The order in which the four synthons1-4 are assembled together is indicated in the bottom row; the numberof carbon atoms in each synthon is written to the left of the synthonnotation (1-4) on the bottom row, where the target molecule is denotedby th e white point.rearrange ments of carbenium ions. In the field of organo-metallic chemistry, Gielen, Brocas, and Willem used GT andother m athematical tools for rationalizing experimental data.IIgNew permutational instruments (double cosets) were developedby Ruch,12' Klemperer,l2I Ugi, and co-workers.122In some instances, the nu mber of isomers is so large thatonly local properties of the reaction graph can be investigated.This is the case of the bullvalene automerization graph,'23which has 10!/3 =1 209 600 vertices if enantiomerism is takeninto account and 604 800 vertices if not.

SYNTHON GRAPHSDeinton's saying "Chemistry is an a rt with a bonus"124 snowhere more valid than in organic synthesis. Woodwardmastered the intricacies of this art in a hitherto unequaledmanner, but C orey and WipkelZs naugurated a new era, whichadds the computer's memory and speed to the chemist's insight.Com puter-aid ed design of orga nic syntheses looks for waysto dissect a molecular graph (th e target molecule) into simple,commercially available starting materials, able to react viaknown reactions to form the target molecule.126 The termsynthon graphs was first used by H en dri ~ks 0n . l~ ' ynthongraphs have vertices symbolizing synthons and edges sym-bolizing bond sets, Le., assemblies between synthons. Anexample is presented in Figu re 1, where th e steroid skeletonis assembled from four synthons; the bond set is denoted bybroken lines and the synthon by solid lines in the upper row.Now adays a rich bibliography exists in this field, and so-phisticated computer programs have been developed and im-plemented.'28-'30 The field is continuously expanding and mayeventually reach the heuristic stag e when new potential re-actions will be uncovered by these computer programs; thistrend was adumbrated more than 20 years ago'31when lists

of electrophiles and nucleophiles were matched matrixwise.I t w as noted on tha t occasion th at whenever protolytic reac-tions were possible, these occurred preferentially convertingthe stron ger acid-base pair into the weaker one before thecoupling of the electrophile with the nucleophile. Thu s,whereas the Gattermann-Koch reaction occurs via formylcation and aromatic compound (and not via carbon monoxideand arenonium cation), the related reaction in the alkene seriesoccurs via carbon monoxide and alkylcarbenium ion (and notvia formyl cation and alkene).GRAPH THEORY AND CHEMICAL INFORMATION

Th e preceding sections contained occasional information oncoding and naming chemical stru ctures. Th e present section


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APPLICATIONSF GRAPH HEORYN CHEMISTRY J . Chem. Znf. Comput. Sci., Vol. 25 , No. 3, 1985 341nection table of the C A S system (see below) and can be usedfor substructure search.

Coding and linear notation systems are older than the nodalnomenclature. In Read's approach,140 he coding of acyclicchemical graphs is achieved by finding the graph center andby writing chemical formulas either a s clusters, delineated byparentheses, attached to the center, or as a linear chain ofclusters. N o special notation is needed because the code usesnormal chemical symbols and, in addition, periods for simplebonds or colons for double bonds in clusters such as ca rbonyl.Th e coding is implemented by a computer program both forcyclic and a cyclic graphs having a co nnectivity table as inp utda ta (atom s and bonds). No sub- or superscripts are used.Thus, 2-methyl- 1-propanol becomes CH( CH3 )2(CH 2.OH ),and malonic acid is CH2(C=O.OH)2.Th e oldest linear notation system was imagined by Gordonand c o-workers,14' but it did not enjoy a widespread use;Dyson's system142 as recommended by IU PA C, but the mostpopular is Wiswesser Line Notation (W LN ).1 43 All thesesystems rely heavily on conventions; e.g., the W L N for hy-droquinone is Q R DR . At present, there exist computerprograms that can convert W LN into connectivity tables, andvice versa.lU Normally, chem ical line notations such as W LNare convenient tools for manual or m achine registration andsubstructure searching of files with mo derate size (104-105structures).The CA S Re gistry File uses connection tables developed byMorgan.'45 Morgan's algorithm is based on the connectivityof each atom in the hydrogen-depleted graph and on he ex-tended connectivity of its adjacent non-hydrogen atoms. Th eextended conne ctivity allows a b etter discrimination betweenatoms than the simple connectivity (which for carbon atomscan have only the four values 1-4).At present, more than 7 million structures are available forCA S online search. Some important features are that one canmake substructure search (manually, such a search may beextremely complicated and time consuming with the CAChemical Substance or Formula Indexes) and that questionscan be compounded with logical symbols (A ND , OR , NO T).An impressive effort was invested in computer programs forgenerating structure diagrams fro m connection tables eitheron a computer screen or on paper (photocomposer or elec-trostatic printer o u t p ~ t ) . ' ~ ~ J ~ ~So far, only constitutional and no stereochemical aspectsw ere taken in to accou nt. Wip ke and D y ~ t t ' ~ ~mprovedMorgan's system so as to include stereochemical information;for implementing Wipke's et al. SECS (Simulation and Eval-uation of Chem ical Synthesis) computer program, one needsfrom the outset all stereochemical nformation about the targe tmolecule and the starting ma terials. It was for this purposethat the SEMA program (StereochemicalExtension of Morgan'sAlgorithm) was developed. A simple approach codes con-figurations at stereocenters (enantiomerism), diastereomerismof double bonds or ring structures, and even conformationsaround single bonds.'48Among other systems for structure storage and retrieval thatare in widespread use, we shall mention Dubois' DARC systembased upon th e idea of focal coding;'49 the In stitute for S ci-e ntific I n f o r m a t i ~ n ' ~ ~akes available this database and alsoIndex Che micus Online. Less widely used systems were de-veloped by Fugmann,15' by Ugi and co-w~rkers , '~~y theAll-Union Institute for Scientific and Technical Informationin Moscow and the Mathematics Institute in Novosibir~k,'~~and by M o r e a ~ . ' ~ ~alaban, Mekenyan, and Bonchev, in aseries of papers,ls5 elaborated hierarchically ordered extendedconnectivities (the H O C algorithm) for coding the constitutionand stereochemistry of molecules, for canonical atom num-bering, and for developing on the basis of this numbering new

will deal exclusively with problems concerning chemical in-formation such a s nomenclature, coding, storage, and retrievalof chemical structures.Chemical nomenclature has grown in a haphazard way,sanctifying trivial names and collecting several systems thatwere widely used, rather than being based upon a logicalfoundation and on first principles. Th e policy of the IU PA CCommission on the Nomenclature of Organic Chemistry andof similar nomenclature commissions for inorganic or mac-romolecular chemistry as well as of the International U nionof Biochemistry is to assemble and simplify the existingpractice of chemical nomenclature, so as to obtain fewer ex-ceptions and a uniform system. However, as stated byFletcher,132 it appears unlikely that the present rules will everbecome systematic." Historical introductions to chemicalnomenclature problems were published by Verkade133 nd byG o ~ d s o n , ' ~ ~nd a b ib liography is a~ ai 1a b l e . l~ ~ne shouldmention that the 1892 International Congress in Genevacreated a Com mission for the Reform of Chemical Nomen-clature and that nowadays inorganic chemistry (red book) andorganic chemistry (blue book) have a set of IU PA C rules'36that are consistent for the most part with the practice ofChemical Abstracts Service (CAS).137A proposal for using graph theory for the purpose ofchemical nomenclature ("nodal nomenclature") was made byLozac'h, Goodson, and P o ~ e l l . ' ~ ~ J ~ ~ead and Miherla hadlisted the attribu tes of an ideal nomenclature sy stem, but theydeveloped actually a coding procedu re (see below). Th e nodalnames13shave a unique numbering of all non-hydrogen andnonsubstituent atoms (nodes) of the graph, un like the classicalnomenclature wherein each moiety (module) has its ownnumbering. Each node is assigned a unique number used inthe desc riptor, i.e., the part of the nam e enclosed in brackets.Modules may be cyclic or acyclic sets of node s, to be treate das sepa rate entities; a descriptor s tartin g with zero belongs toa cyclic module; otherw ise, the module is acyclic. A fewexamples will be analyzed. n-Hexane is called [6]hexanodane,while 3-methylpentane s called [5.13]hexanodane. In the lattercase, the sum of numbers inside the brackets is the numberof carbon atom s, Le., 6, and the upper index 3 indicates whichof the five atoms of the ma in cha in is connected to the one-atom side chain. Bicyclo[2.2.2]hexane is called bicyclo-[06.21,4]octanodane, nd indane is called bicyc10[09.0'~~]no-n a n ~ d a n e . ' ~ ~ ~ ' ~ ~By means of the set of seniority rules, and of several ex-a m p l e ~ , ' ~ ~t is quite easy to become fam iliar with the nodalnomenclature. The advantages of this system are its generality,the consistent treatment of rings, chains, or combinationsthereof, systematic numbering of atoms, possibility of appli-cation to coordination comp ounds and to cyclophanes, and itscomputer compatibility.The full nodal name of 50 is 14-chloro- 10-oxo-2-aza- 17-oxatricyclo [ 09.0'*')2: 1O( 1)10: 1 1 06)7: 17(2)3: 19( 1)4:20-(l)]icosan( 1,3-9,11-16)axene-2-carboxyliccid. The modules

50are separated by colons, and numbers outside parentheses arelocants. Th e double bonds are indicated individually (e.g., 1)or collectively in conjugated system s (e.g., 3-9, or 11-16) asa prefix to the word axen e; the -CO OH, -C1, or =O groupas well as heteroatoms is indicated as replacement terms. Thedescriptor from the nodal name is comparable with the con-


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342 J . Chem. If. Comput.Sci., Vol. 25 , N o. 3, 1985topological indices a unique topological representation, cor-relations with H NM R spect ra in PAHs, and computerprograms for the implementation of this algorithm; it improvesthe M organ algorithm and enables more compact storage ofstructure information in the computer memory. A somewhatsimilar but less extensive approach for inorganic structuressuch as polyhedral clusters, including a unique linear notation,was proposed by Herndon and co-worker~.~~

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