Solved and Unsolved Problems in Chemical Graph Theory

Quo Vadis, Graph Theory? J. Girnbel, J.W. Kennedy & L.V. Quintas (eds.) Annals of Discrete Mathematics, 55, 109-126 (1993) 0 1993 Elsevier Science Publishers B.V. All rights reserved.

SOLVED AND UNSOLVED PROBLEMS IN CHEMICAL GRAPH THEORY

Alexandru T. BALABAN Department of Organic Chemistry, Polytechnic Institute Splaiul Independentei, Bucharest, ROUMANIA

Abstract Chemistry and graph theory meet in several areas which are briefly reviewed. A few solved and unsolved problems are discussed generalized centers in cyclic graphs; irreducible sequences in polymers; cages; spectral graph theoretical problems; k-factorable graphs with k > 1, and perfect matchings with k = 1.

1.

Introduction

Points of contact between graph theory and chemistry exist from the very beginning of graph theory. It is an established fact that the birth of graph-theory occurred from three independent areas: mathematics via Euler’s famous problem of the seven bridges in Konigsberg; electricity via Kirchhoffs electrical network theory; and chemistry via Cayley’s enumeration of alkane isomers. The latter problem continued to attract both mathematicians and chemists, and led to P6lya’s celebrated enumeration theorem. Even Sylvester, who coined the name “graph”, was fascinated by the theory of chemical structure in organic chemistry, due mainly to KekulC. A brief review entitled “Early History of the Interplay between Graph Theory and Chemistry” was published as the first chapter of a monograph Chemical Applications of Graph Theory [11. At present, chemistry and graph theory are expanding very rapidly because both are faced with challenging problems, and in both cases the unknown peaks lie close at hand, needing fewer approaching expeditions and base camps than other scientific disciplines. By cross fertilization, during the last 30 years, the interdisciplinary areas of graph-theoretical applications in chemistry has become a recognized research field with its own journals [2] [3], monographs and symposia [4]-[ 111. Describing my own experience in this field, after organic chemistry, I registered as a student in mathematics. However, I could not finish the latter studies because a third opportunity (of “once only” type) arose for a one-year training program in nuclear physics and radiochemistry. Thus, as a fresh Ph.D., in 1959 I published my first chemistry papers. In my first article I tried to solve a graph-theoretical problem, connected with the enumeration of all possible monocyclic aromatic systems [12]. It was actually the “necklace problem” with restrictions as to adjacencies, and it had several sequels [13] [14]. From the outset, two facts became clear: (i) in order to work in a borderline area, one has to be familiar with all relevant disciplines; (ii) the best results are obtained by cooperation between specialists, provided they bridge the barriers of terminology and publication style characteristics peculiar to each discipline. At present, among the areas of intensive activity in graph-theoretical applications in chemistry, one may cite: (a)

Quantitative structure-property (or activity) relationships (QSPR or QSAR), especially for drug design, using topological indices;

(b)

Reaction networks, including retro-fragmentations for the design of organic syntheses;

110

(c)

A.T. Balaban

Coding and nomenclature of chemical structures, including input and retrieval of chemical information for documentation purposes.

All of these areas have reached the stage of providing successful commercial services. Some of the computer programs are commercially available. The main idea behind most of the graph-theoretical applications in chemistry is the oneto-one correspondence between chemical structures and their constitutional graphs, wherein atoms are represented by vertices, and covalent bonds by edges. I t is customary in organic chemistry to use hydrogen-depleted graphs, where only non-hydrogen atoms are indicated by vertices. Chemical structures are discrete entities, whereas their properties vary continuously. Graph theory aims at unique representation (alphanumerically), coding, ordering, and enumeration, of all possible chemical structures. As at 1990, more than ten million compounds have been reported in the chemical literature, and at the present rate of growth, this number will double in about 20 years. Molecular formula (such as C12H22011for sugar) can be easily manipulated, ordered, and retrieved. However, the huge number of isomers (that is, substances having the same molecular formula and differing in their structure due to differences in the topological or geometrical mode of atom bonding) complicates the situation and requires help from graph theorists. In retrieving information, chemistry can manipulate structures which are represented by graphs without the need of using words, since words are often ambiguous or imprecise. Direct access to Chemical Abstracts, Beilstein, or Gmelin databases allows a chemist to learn in a few minutes whether a given structure or substructure has been described in the last few decades. In the near future, all lo7 compounds will be included in these databases. Thus for most purposes, namely those where structural formulas rather than words are involved, chemists can safely assert that chemistry is the best documented science. Physical chemistry, or chemical engineering, which need words or keywords, are only as well documented as mathematics, physics, medicine, or law. The present paper will present a few solved or unsolved problems involving chemical graphs, in the hope that the latter problems will serve as challenge and incentive for graph theorists. It should be made clear that the choice of topics is subjective and is linked to personal interests rather than to the intrinsic importance of the problems . 2.

Graph Centers, Chemical Nomenclature and Documentation

Traditional chemical nomenclature and documentation is based upon the system adopted by the International Union of Pure and Applied Chemistry (IUPAC) which seeks in acyclic structures the longest linear chain. When several such chains are present, there exist hierarchical rules for making a unique choice. This chain is numbered starting from one end; again, the choice of the canonical end is governed by elaborate rules. It is easy to see that this system is cumbersome. Still more intricate are the rules for the IUPAC nomenclature of polycyclic systems. Significant progress was achieved on a graph-theoretical basis in the proposed nodane nomenclature which, however, is still based on the longest chain [15][16]. The enumeration of 4-trees (trees with vertex degree at most four) in terms of both the number of vertices (carbon atoms in hydrocarbons) and the number of vertices in the longest chain was effected by means of Pdya's theorem, using a specially devised computer program [ 171. The unique and simple graph center or centroid of any tree has led Read to devise a centric representation for acyclic chemical compounds [18]. He has also proposed an extension of these ideas to cyclic chemical systems [19]. In cooperation with Bonchev, Mekenyan and RandiC [20] [21] we

Solved and unsolved problems in chemical graph theory

111

have developed an algorithm for finding the generalized center for any graph. Briefly, the algorithm consists in finding sequentially: A.

(i) minimum vertex eccentricity; (ii) minimum vertex distance sum; (iii) minimum number of Occurrences of the largest distance (or, when this is the same for two or more vertices, the next largest distance, etc.);

B.

The same parameters as in A.(i)-(iii) but for edges.

Thus, vertices and edges are ranked into equivalence classes (the most central vertices/ edges have the smallest ranks). When the same rank sum is obtained from different summands, priority is given to the partition containing the smallest rank. Then the above steps are iterated replacing the word distance by rank, until the ranlung of vertices and edges undergoes no further modification. The center consists of the vertex/vertices with lowest rank. Although this idea reduces substantially the number of central vertices in polycyclic graphs, allowing in principle the simplification of chemical coding or naming by analogy with Read's approach, we feel that the last word in this respect has not yet been said.

3.

Irreducible Sequences in Polymers

Polymers (natural or synthetic) are essential for life (for example, proteins, nucleic acids) and for civilized life (for example, plastics, composites, elastomers). In stereoregular polymers, the three-dimensional configuration of chiral atoms can lead to various types of sequences which may be detected experimentally by nuclear magnetic resonance or by the thennal/mechanical properties. In isotactic polypropylene all configurations are of the same kind (. . .RRR.. .) and this polymer has higher strength and melting point than irregular (atacTable 1: Numbers of necklaces ( N K ) and of irreducible sequences ( I S ) for each partition R,S, ...U , ( r + s + ... + u = m ) , total numbers N( m,n) of irreducible sequence for given numbers of n of comonomers and m of mers in the repeating irreducible sequence.

I

n=2

n=4

n=3

N K Partition IS N(m,n) N K Partition IS N(m,n) NK Partition -

-

-

-

1 RS

1

1

1 R2S

1

1

1 RST

1

1

1 R3S 2 R2S2

1

2

2 R2ST

2

2

2 R3ST 4 R2S2T

2 3

3 R4ST 6 R3S2T 1 1 R2S2T2

3 6 4

3 RsST R4S2T 10 R3S3T 18 R3S2T2

3

1 R4S 2 R3S2

1 2

1 R5S 3 R4S2 3 R3S3

1 2 2

1 RgS 3

3

~ 5 5 2 D C

4 R4S3

1 1 J

4

5

YI Q

-

7 12

-

IS N(m,n) -

3 RSTU

-

1

1

6 R2STU

2

2

13

10 R3STU 16 R2S2TU

3 8

11

21

15 R4STU 4 30 R3S2TU 17 48 R2S2T2U 12

33

~

A.T. Balaban

112

tic), polypropylene. Alternating configurations (. ..RSRSRS.. .) are encountered in syndiotuc-

tic polymers. In binary copolymers the two comonomers can also give rise to various sequences. Let the comonomers in higher copolymers (ternary, quaternary, etc.,) be denoted by R, S, T, U. All irreducible sequences of these comonomers, whose infinite repetition leads to a polymer chain, have been enumerated [22]-[24]The basic idea is to start from the necklace problem and to eliminate those necklaces (with rn beads of n colors) which on opening and linking into an infinite chain are reducible to smaller necklaces. Table 1 and Table2 present the numbers of irreducible sequences as well as the sequences themselves for the simplest cases.

Table 2: Irreducible sequences with n = 2 , 3 or 4 (binary, ternary or quaternary copolymers) and sequence lengths rn = 2 through 7. n=2

'1

I

-

UlRS

RRSS

lRRRS

RRRSS RRSRS

MRRS

n =4

n=3

RS

!

RRST RSRT RRRST RRSRT

RSTU RRSST RRSTS RSRST

RRRRSS RRRSSS RRRRST RRRSST RRSSTT RRRSRS RRSRSS RRRSRT RRRSTS RRSTST RRSRRT RRSRST RRSTTS RRSRTS RSRTST RRSRTT RSRSRT

M R R R S RRRRRSS RRRRSSS RRRRRST RRRRSST RRRRSRS RRRSRSS RRRRSRT RRRRSTS RRRSRRS RRSRRSS RRRSRRT RRRSRST RRSRSRS RRRSRTS RRRSRTT RRSRRST RRSRRTT RRSRSRT RRSRTRS

-

RRSTU RSRTU RRRSTU RRSRTU RSRTRU

RRRSSST RRRSSTT RRRRSTU RRRSSTU RRRSSTS RRRSTST RRRSRTU RRRSTSU RRSRSST RRRSTTS RRSRRTU RRRSlTU RRSRSTS RRSRSTT RRSRTRU RRRSTUS RRSRTSS RRSRTST RRSRSTU RRSSRTS RRSRTTS RRSRTSU RSRSRST RRSSRTT RRSRTTU RRSTRST RRSRTUS RRSTRTS RRSRTUT RRSTSTS RRSRTUU RSRSTRT RRSSRTU RSRTRST RRSTRSU RRSTRTU RRSTRUS RSRSRTU RSRSTRU RSRTRSU

RRSSTU RRSTSU RRSTTU RRSTUS RSRSTU RSRTSU RSRTUT RSTRSU RRSSTTL RRSSTU? RRSTSTU RRSTSUS RRSTSU? RRSrrSL RRSTUST RRSTUTS RSRSTUT RSRTSTU RSRTSLTI RSTRSTU

4. Cages and Reaction Graphs Unlike the constitutional (molecular) graphs discussed so far, in which vertices symbolize atoms and edges symbolize covalent bonds, in the graphs about to be discussed a vertex represents a molecule or a reactive intermediate, and an edge represents an elementary reaction step. Such graphs are termed reaction graphs. Two isomorphic graphs can result from quite different chemical contexts:

Solved and unsolved problems in chemical graph t h e ~ r y

113

(i) Rearrangements of carbocations (Scheme l).The scheme depicts the two (unordered) substituents linked to the positively charged carbon atom, and this can be either at the left or right of the C-C bond symbolized by a period [25]-[27l. The reaction step involves the shift of a substituent from the vertex of degree four to that of degree three.

Scheme 1: Portion of the reaction graph for rearrangements of carbocations. (ii) Pseudorotation of pentacoordinated compounds. This is exemplified by phosphoranes with pentavalent phosphorus at the center of the trigonal bipyramid [28] [29] (Scheme 1). This

4

;5-2

5 (.451

(23.) 4

(13.)

4 (12.1

Scheme 2 Portion of the reaction graph for pseudorotation of trigonalbipyramidal compounds. scheme indicates the two (unordered) apical substituents, and the period discriminates among the two resulting enantiomers. The reaction step involves the conversion of the two apical substituents mutually situated at an angle of 180"into equatorial substituents at an angle of 120";the third equatorial substituent of the new configuration stays fixed during the rearrangement (pivot substituent); the remaining two formerly equatorial substituents become the new apical substituents by increasing their angle from 120"to 180". The resulting graph is bipartite, regular of degree three (cubic graph), has 20 vertices, and is known as the Desargues-Levi graph (see Figure 1).

114

A.T. Balaban

2L 35@ :3

14

25

Figure 1: Reaction graph corresponding to Scheme 1 and Scheme 1 (Desargues-Levi graph with 20 vertices), and the Petersen graph with 10 vertices (5-cage). If in (i) the two carbon atoms are indistinguishable (no isotopic label), or if in (ii) enantiomerism is ignored, the period in the above notation vanishes and, by painvise identification of the antipodes, the above graph reduces to the 5-cage (Petersen graph) with 10 vertices 1251 [30]. A g-cage or (3,g)-cage is defined to be a cubic (trivalent) graphs with girth g, having the smallest number of vertices [31]-[33]. On examining the known cages it is evident that they are related to each other (see Figure 2). For the cages with odd girth only one representation with a (g + 1)-circuit is shown, but for those with even girth two representations with g- and (g + 2)-circuits are presented[34]. It is easy to see how, on excising trees (shaded area in Figure 2) from the even g cages in the representation with (g + 2)-circuits one obtains the cage with girth equal t o g - 1. Table 3 shows the conjectured excised trees. One exception is the 9cage. At this time, Biggs [35], Evans [36], and McKay [37], have found eighteen such 9-cages having low symmetries with 58 vertices. The excision procedure leads to (3,9)-graphs with 60 vertices, starting with one of the three known 10-cages [38] [39] with 70 vertices, as is shown in Figure 3 [38]. The same procedure, applied to the unique known 12-cage (or Benson graph) [a], leads to the conjectured (uniquely so at this time) 11-cage with 112 vertices [34], shown in Figure 4 and Figure 5. Nothing is known about cages with girth higher than 12; the low symmetry of the (39)-graphs with 58 vertices raises the question if lower numbers of vertices might perhaps lead to higher symmetries in this case. A challenge for programmers would be to devise a computer game which would highlight the high symmetry of most cages. Table 4 shows the girth g and order n of the known trivalent cages. On comparing the numbers of automorphisms of the tetrahedron (namely, 12 automorphisms) and the 3-cage (24 automorphisms), both having n = 4 vertices, it is evident that the symmetry operations for the graph are much more numerous than for the corresponding polyhedron. It would be interesting to see on the screen (in the game), by various edge colorings, all edge and s-path automorphisms of the Petersen graph (5-cage) or the Tutte graph (8-cage); the former graph has 120 automorphisms and is 3-regular (3-unitransitive), while the latter has 1440 automorphisms and is 5-regular. 5.

Spectral Graph Theory In the well-known Hiickel molecular orbital (HMO) theory, the eigenvalues of graphs

Solved and unsolved problems in chemical graph theory

115

g= 3 4

5

?

7

6

7

8

7

Figure 2: Representation of cages by bridging opposite vertices in circuits with paths. Excision of shaded trees converts a g-cage with even g into a (g - 1)-cage. whose vertex degrees are at most three are of three types: negative, representing bonding nmolecular orbitals (BMOs); positive, representing anti-bonding n-MO's (ABMO's); and zero, corresponding to non-bonding n-MO's (NBMO's). Normally, for most cyclic or acyclic molecules having an even number of carbon atoms in conjugated systems, the number of BMO's equals that of ABMOs, and there is no NBMO. The homodiatomic triple-bonded nitrogen molecule (N2) has a very high stability because all BMO's are filled with electrons, there is no NBMO, and all ABMO's are vacant. Exactly

116

A.T. Balaban

Table 3: Excised trees from g-cages with even g, for converting them into (g - 1)-cages.

the same situation and hence stability occurs in aromatic molecules such as benzene with 4 k + 2 7c-electrons, where k = 0, 1,2, ... . In polycyclic molecules having delocalized 7c-electrons, various situations may occur. One of the most peculiar and challenging is to have no NBMO's, and to have more positive than negative eigenvalues, or vice versa, as it was pointed out first by Bochvar and Stankevich. Several examples are gathered below, but the general rules (that is, necessary and sufficient conditions) are not yet clear. Two classes exist [41]: Class A : Graphs with an excess of negative eigenvalues over positive ones: 2j pairs (j= 1,2, ...) of ( 4 k + 1) -membered rings condensed (that is, sharing one edge) directly, or via one (4k + 2 ) -membered ring, or via two 4k-membered rings, in a centro-symmetrical arrangement. Examples are shown in Figure 6. Class B: Graphs with an excess of positive eigenvalues over negative ones. Similar to class A , but with 2j pairs of ( 4 k + 3) -membered rings. Examples are presented in Figure 7.

6.

k-Factorable Graphs with k > 1

Decomposition of graphs into congruent factors has interesting chemical implications, the most important of which will be described in the next section. Here we shall discuss a less studied application, namely decomposition into factors with at least three vertices. Terpenoids are naturally occurring compounds having a polyisoprenic skeleton, corresponding to factors with five vertices in a branched chain having a vertex of degree three. Living cells synthesize terpenoids via the reaction of acetyl-coenzyme A( 1) with acetoacetyl-coenzyme A(2) which

Solved and unsolved problems in chemical graph theory

117

Figure 3: Two representations for one of the three 10-cages [38]. affords mevalonic acid ( 3 ) .This is phosphorylated and decarboxylated yielding geraniol pyrophosphate 4 (the pyrophosphate unit is symbolized by OPP), which by sequences of reactions leads to acyclic compounds such as farnesol 5 or rubber 6, while by cyclization squalene 7 yields cholesterol 8 and its derivatives (see Scheme 3). Other isoprenoid graphs are shown in Scheme 4: monoterpenes such as pinene 9, paracymene 10, camphor 11; sesquiterpenes: guajazulene 12, vetivazulene 13; diterpenes such as

118

A.T. Balaban

Figure 4: Derivation of the conjectured 11-cage from Benson's 12-cage by excising a tree with 14 vertices. retinol (vitamin A) 14. In all above cases the isoprene unit (factor) is shown with full lines, and dotted lines link these units. In all terpenoids, the molecular graph is decomposable into congruent factors. The problem in chemistry is to detect whether a given graph is factorable into similar isoprenoid factors, and vice versa to generate such polyisoprenoid graphs. In collaboration with Professor S. Marcus from the Faculty of Mathematics of Bucharest University, by using picture grammars and push-down automata, several computer programs were devised for this purpose [42][44].It would be interesting to apply other methods to this problem, and to generalize the problem for other k-factors.

7. Perfect Matchings (Factorable Graphs) In chemistry, molecular graphs that can be decomposed into 1-factors (K2 graphs) have a special significance, especially for polyhex graphs. Such graphs represent polycyclic aromatic hydrocarbons (PAH's) and they have higher stability when they are 1-factorable than when

Solved and unsolved problems in chemical graph theory

119

Figure 5: The conjectured 11-cage. Inner vertices, having other vertices at distance eight, belong to a different orbit from the outer vertices [34].

~

g

3

4

5

6

7

8

9 a 1 0 b 1 1

12

n

4

6

10

14

24

30

58

126

70

112

they are not, or when they have more such factorizations (also called perfect matching, or KekulC structure counts) [45] [&I. A necessary but insufficient condition for the graph to have at least one Kekule structure is that it has an even number of vertices. Recently, a set of necessary and sufficient rules for polyhexes to be I-factorable was published [43. Examples of even-numbered polyhexes which have no 1-factorization(called concealed non-Kekulhzn) 15 and 16, are presented in Scheme 5. Polyhexes (PAH's or benzenoids) are of three types: catafusenes, penfusenes, and coronafusenes (coronoids). As indicated by Balaban and Harary [ 4 7 , the dualist (characteristic)

I20

A.T. Balaban

Qm

Figure 6:Examples of class A graphs.

Figure 7:Examples of class B graphs. graph is a useful criterion for discriminating among these three types. Its vertices are the centers of hexagons and its edges connect condensed hexagons (that is, hexagons sharing two adjacent vertices, representing two carbon atoms). Unlike graphs, the angle between edges of dualist graphs is important. Unlike dual graphs, in dualist graphs there is no vertex corresponding to the outer region. The dualist graphs of catafusenes are trees; those of perifusenes

121

Solved and unsolved problems in chemical graph theory

~CH~CO-SCOA 4 CH3COCH2CO-SCOA

1

1 +2

--*

2

HO I HO-CHz-CH2-C-CH2-CmH I

3

4

CH3

OPP

4

8

Scheme 3: Examples of terpenoids (polyisoprenoid graphs) and their biosynthesis.

Scheme 4: Examples of polyisoprenoid graphs. have 3-membered rings; those of coronoids have larger rings which are not the periphery of assemblies of 3-membered rings. Several coding systems have been devised on the basis of dualist graphs for polyhexes. Cata-condensed appendages may be present in peri- and coronafusenes, and pen-fused subgraphs may be present in coronoids [@] [49]. Examples are presented in Scheme 6.

122

A.T. Balaban

15

16

Scheme 5: Examples of non-KekulCan perifusenes with even number of vertices.

Scheme 6: Examples of polyhexes with their dualist graphs: chrysene (catafusene), the carcinogenic benzopyrene (perifusene), and Kekulene (coronafusene). One may consider most polyhexes as portions of the graphite lattice; however, this is not always true. Indeed, polyhexes may or may not be embeddable in a plane without vertices coinciding. An example of the latter is 7-helicene shown as the last polyhex in Scheme 6, which is an out-of-plane catafusene. In work with Tomescu we defined isoarithmic polyhexes as PAH's which have the same numbers of hexagons and the same K values (see next paragraph), but differ in their topology. For example, 17 and 18 (see Scheme 7). Also, we applied algebraic methods for enumerating K values of catafusenes obeying certain composition rules [MI-[531.

17

18

19

Scheme 7: Isoarithmic catafusenes (17 and 18), an acene (19) and their dualist graphs. The number K of perfect matchings (1-factorizations, or KekulC structures) plays an important part in the so-called valence bond theory of PAH's, and an appreciable part of theoretical chemical papers is devoted to such topics. For example, the n-acenes 19 having K = n + 1 are less stable than n-helicenes 17 or other isoarithmic systems like the zig-zag catafusenes 18; in general, such fibonacenes as 17 or 18 have K = F , where F , is the n-th Fibonacci number.

Solved and unsolved problems in chemical graph theory

123

In Erich Huckel's MO theory of aromatic character (which refers to electronic delocalization and stability, and not to smell) the molecular orbitals (MO's) for the x-electrons (one for each carbon atom in benzenoids) are found with the help of the adjacency matrix. From it one obtains the characteristic polynomial whose roots xi = ( a- Ei)/p afford the orbital energies E , in bunits relative to a value a (Coulomb integral). Thus for benzene (CH)6 also called annulene, all bonding levels (BMO's) are occupied, there is no NBMO, and all ABMO's are vacant, resulting in a closed n-electron shell, or x-electron sextet (the arrows in Scheme 8 indicate n-electrons with their spin). In a far-reaching generalization, Huckel showed that molecules with 4k + 2 melectrons in a delocalized system have aromatic character [%I[ 5 7 . Examples are shown in Scheme9: benzene 20, naphthalene 21, phenanthrene 22, anthracene 23, azulene 24, cyclopentadiene anion 25, tropylium cation 26, thiophene 27, 18annulene 28, and tetra- t-butyl-bis-dehydro- 14-annulene 29. Energy

..:Dl

ABMO's

a+p 4 a

a-2P

1

fc

Scheme 8: Molecular orbitals of benzene (Qannulene). An unsolved problem is the following: what are the general structural patterns for in-plane (and separately for out-of-plane) polyhexes with maximal K values for any given number h of hexagons in the polyhex? A brute-force approach led to the following conjecture: the polyhex is a branched catafusene; for out-of-plane catafusenes and for certain h values (4,10,22,46, .. .), Gutman [58]pointed out the most branched structures. For in-plane catafusenes, Table 5 presents the dualist graphs with h I 13 and the corresponding K values. The two cases with asterisks (h = 11 and 12) have two isoarithmic solutions each, and give higher K values for corresponding out-of-plane catafusenes (305and 510, respectively). The cases with h = 9 and 13 have isoarithmic out-of-plane catafusenes.

Note Added in Proof: The last problem was recently solved for out-of-plane benzenoids. The corresponding problem for in-plane benzenoids is still unsolved.

124

A.T. Balaban

20

21

22

'I

\

23

27

26

24

Scheme 9: Examples of aromatic molecules obeying Huckel's 4k + 2 n-electron rule (each double bond or heteroatom contributes two x-electrons); the numbers of n-electrons are inscribed in the formule. Table 5: Dualist graphs of the in-plane catafusenes (with h hexagons) possessing the highest numbers K of KekulC structures. h

K

h

K

1

2

2

3

5

4

9

4

h

K

3

5

14

6

24

H

7

41

8

66

9

110

10

189

11

302

12

504

13

863

9

References [I]

A.T. Balaban and F. Harary;Early history of the interplay between graph theory and chemistry, in Chemical Applications of Graph Theory, A.T. Balaban (editor), Academic Press, London-New York, 1 4

[2]

P.G. Mezey and N. Trinajstii (editors); Journal of Mathematical Chemistry, Balzer Publ., Basel, 1 (1987).

(1976).

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A.T. Balaban, A. Dreiding, A. Kerber and O.E. Polansky (editors); Mathematical Chemistry, Mulheiml Ruhr, l(1975). N. TMajstiC; Chemical Graph Theory, 2nd. edition, CRC Press, Boca Raton, Florida (1992). R.B. King (editor); Chemical Applications of Topology and Graph Theory, Elsevier. Amsterdam (1983). R.B. King and D.H. Rouvray (editors); Graph Theory and Topology in Chemistry, Elsevier, Amsterdam (1987). m D.H. Rouvray (editor); Computational Chemical Graph Theory, Proceedings of the 1988 American Chemical Society Meeting in L o s Angeles, Nova Science Publ. Inc., New York (1989). J.W. Kennedy and L.V. Quintas (editors); Applications of Graphs in Chemistry and Physics, Nod-Holland, Amsterdam (1988). R.C. Lacher (editor); MATHICHEWCOMP 1987, Elsevier, Amsterdam (1988). D.H. Rouvray and A.T. Balaban; Chemical applications of graph theory. in Applications of Graph Theory (R.J. Wilson and L.W. Beineke, editors), Academic Press, London, 177-221 (1979). A.T. Balaban; Applications of graph theory in chemistry; J. Chem. In& Comput. Sci., 25.334-343 (1985). A.T. Balaban; An attempt towards the systematics of monocyclic aromatic compounds, Studii Cercet. Chim. Acad., Romania, 6,257-295 (1959).(Roumanian). A.T. Balaban and F. Harary; Chemical graphs: IV. Dihedral p u p s and monocyclic aromatic compounds, Rev. Roumaine Chim., 12,1511-1515( 1%7). K.Lloyd; The footballers of Croam, London Math. SOC.Lecture Notes Series, 13.97-102 (1974). A.L. Goodson; Graph-based chemical nomenclature. I. Historical background and discussion, J. Chem. In$ Compur. Sci., 20, 167-172 (1980). A.L. Goodson; Graph-based chemical nomenclature.11. Incorporationof graph-theoretical princlples into Taylor’snomenclature, J. Chem. InJ Comput. Sci., 20. 172-176 (1980). A.T. Balaban, J.W. Kennedy and L.V. Quintas;The numher of alkanes having n carbons and a longest chain of length d: An application of a theorem of P6lya. J. Chem. Educ., 65.3W3 13 (1988). R.C. Read; The coding of trees and tree-like graphs, University of the West Indies, Jamaica (1968).preprint. R.C. Read and R.S. Milner; A new system for the designation of chemical compounds for the purpose of data retrieval. 11. Cyclic compounds, Report to the University of West Indies, Jamaica (1968). D. Bonchev, A.T. Balaban and M. RandiC; The graph center concept for pdycyclic graphs, Int. J. Quanrum Chem., 19,6142(1981). D.Bonchev, 0.Mekenyaa and A.T. Balaban; Iterative procedure for the generalized graph center in polycyclic graphs, J. Chem. In$ Compul. Sci., 29,9147(1989). A.T. Balaban and C. Artemi;Mathematicalmodeling of polymers.I. Enumeration of non-redundant (irreducible) repeating sequences in stereoregular polymers, elastomers, or in binary copolymers, Math. Chem., 22,3-32 (1987). 1231 C. Artemi and A.T. Balaban; Mathematical modeling of polymers. 11. Irreducible sequences in n-ary copolymers, Math. Chem., 22.77-100 (1987). A.T. Balaban and C. Artemi; Mathematical modelling of polymers. 111. Enumeration and generation of “I repeating irreducible sequences in linear bi-, ter-, qnater-, and quinquenary copolymers and in stereoreplar homopolymers,Makromol. Chem.,189.863470 (1988). A.T. Balaban, D. Farcasiu and R. Banica; Chemical graphs: Part 2.Graphs of multiple 1.2-shiftsin carbonium ions and related systems, Rev. Roumaine Chim., 11,1205-1227(1966). A.T. Balaban; Chemical graphs: Part 16.Intramolecularisomerization of octahedral complexes with six different ligands, Rev. Roumaine Chim., 18,841-854(1973). A.T. Balaban; Chemical graphs: Part 19.Intramolecularisomerization of trigonal-bipyramidal structures with five different ligands, Rev. Roumaine Chim., 18,855-862 (1973). P.C. Lauterbur and F. Ramirez; Pseudorotation in trigonal-bipyramidalmolecules, J. Amer. Chem. SOC., 90,6722-6726(1968). K.E. DeBruin, K. Naumann, G. Zon and K. Mislow; Topological representation of the stereochemistry of displacement reactions at phosphorus in phosphonium salts and cognate systems, J. Amer. Chem. SOC., 91,7031-7040(1%9). J.D. Dunitz and V. Prelog; Ligand reorganization in the higonal bipyramid, Angew. Chem. Internat. Ed. Engl., 7,725-726 (1968).

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W.T. Tutte, Connectivity in Graphs, University of Toronto Press (1966). P.K. Wong; Cages - a survey, J. Graph Theory, 6, 1-22 (1982). F. Harary; Graph Theory, Addison-Wesley, Reading, Mass., 174 (1959). A.T. Balaban; Trivalent graphs of girth nine and eleven and relationships between cages, Rev. Roum. Math. Pures Appl., 18, 1033-1043 (1973). I351 N.L. Biggs and M.3. Hoare; A trivalent graph with 58 vertices and girth 9, Discrete Math., uf,299-301 (1980). I361 C.W. Evans; A second graph with 58 vertices and girth 9, J. Graph Theory, 8.97-99 (1984). [37l B. McKay; - Personal communication. 1381 A.T. Balaban; A trivalent graph of girth ten, J . Comb. Theory, Ser. B, 12, 1-5 (1972). PSI M. O'Keefe and P.K. Wong; A smallest graph of girth 10 and valency 3, J. Graph Theory, 5 , 7 9 4 5 (1981). C.T. Benson; Minimal regular graphs of guzh eight and twelve, Cmud. J. Math., 18, 1 0 9 - 1 0 9 4 (1966). A.T. Balaban; Chemical graphs: Part 17. cata-condensedpolycyclic hydrocarbons which fulfil Huckel's rule but lack closed electmnic shells, Rev. Roumaine Chim., 17, 1.531-1543 (1972). [421 A.T. Balaban, M. Barasch and S. Marcus; Computer program for the recognition of acyclic regular isoprenoid structures, Math. Chem., 5,239-261 (1979). A.T. Balaban, M. Barasch and S. Marcus; Picture grammars in chemistry. Generation of acyclic iso[431 prenoid structures, Math. Chem., 8, 193-213 (1980). [441 A.T. Balaban, M. Barasch and S.Marcus; Computer program for the recognition of standard isoprenoid structures. Math. Chem., 8,215268 (1980). r45l S.J. Cyvin and I. Gutman; Kekule' Structures in Benzenoid Hydrocarbons, Lecture Notes in Chemistry, #46, Springer, Berlin (1988). 1461 J.R. Dias;Handbook of Polycyclic Hydrocarbons, Elsevier, Amsterdam (1987). I47l A.T. Balaban and F. Harary; Chemical graphs: Part 5. Enumeration and proposed nomenclature of benzenoid cam-condensed polycyclic aromatic hydrocarbons, Tetrahedron, 24,2505-2516 (1968). I481 A.T. Balaban; Chemical graphs: Part 7. Proposed nomenclature of branched cata-condensed benzenoid hydrocarbons, Tetrahedron, 25,2949-2956 (1%9). A.T.Balaban; Challenging problems involving benzenoid polycyclics and related systems, Pure Appl. Chem., 54, 1075-1096 (1982). A.T. Balaban and I. Tomescu; Chemical graphs: Part 41. Numbers of conjugated circuits and Kekulk structures for zigzag catafusenes and (j.k)-hexes; generalized Fibonacci nu&&, Math. Chem., 17,91120 (1%). A.T. Balaban, C. Artemi and C. Tomescu; Algebraic expressions for Kekulk structure counts in nonbranched regularly cam-condensed benzenoid hydrocarbons, Math. Chem., 22.77-100 (1987). A.T. Balaban and I. Tomescu; Alternating 6-cycles in perfect matchings of graphs representing condensed benzenoid hydrocarbons,Discrete Appl. Mdh., 19.6-16 (1988). (Reprinted in [8]). 1531 I. Tomescu and A.T. Balaban; Decomposition theorems for calculating the number of Kekulk structures in coronoids fused viapennaphthenyl units, Math. Chem.. 24.289-309 (1989). I541 A. Streitwieser; Mofecular Orbital Theory for Organic Chemists, Wiley, New York (l%l). ~ 5 1 C.A. Coulson, B. O'Leary and R.B. Mallion; Hiickel Theoryfor Organic Chemists, Academic Press, London (1978). 1561 E. Heilbronner and H. Bock; Das HMO-Model1 und seine Anwendung, Verlag Chemie, Weinheim (1%). [TI A.T. Balaban, M. Banciu and V. Ciorba; Annulenes, Benzo-, Hetero-, Homo-Derivatives and Their Valence homers, CRC Press, Boca Raton, Florida (1987). I. Gutman; A class of benzenoid systems with large number of Kekulk structures, J. Serb. Chem. SOC., 53, 607-612 (1988).