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Journal of Mathematical Chemistry 11(1992)65-78

65

RING-TRANSFORMATION GRAPHS AND THEIR APPLICATION TO DEGENERATE HETEROCYCLIC REARRANGEMENTS

Eugeny V. BABAEV and Nikolai S. ZEFIROV Department of Chemistry, State University Moscow, 119899 Moscow, Russia

Abstract
A special type of reaction graphs - the ring-bond-redistribution graphs - is used as a new model for description, classification and enumeration of ring transformation reactions of heterocycles, particularly of the class of degenerated heterocyclic rearrangements. Combinatorial formulae for enumeration of all the possible degenerated rearrangements for heterocycles with only one heteroatom are proposed.

1.

Introduction

Heterocycles form one of the most important and well-investigated classes of organic molecules due to their occurrence in the human body and wide spectra of biological activity. One characteristic of the chemistry of heterocycles consists in the reactions of ring transformation or recyclization, i.e. easy opening of the starting heterocyclic ring and easy closure into a new (or the same) one. This ability of heterocyclic compounds was discovered more than a century ago and caused the now highly developed and extensively reviewed branch of organic chemistry [1-10]. Because this type of reaction is independent of the nature and size of the heterocyclic ring, it is a useful tool in planning synthesis of target organic compounds. Nevertheless, the general picture of heterocyclic ring transformations is rather complicated: they are poorly classified, and there is still not any appropriate mathematical model capable of rationalizing this interesting and beautiful class of organic reactions. It seems that the first (and unique) attempt to apply the methodology of graph theory to this subject was an approach by Balaban [5,7], where labelled subgraphs of the pyrillium ring (incorporated into a new heterocyclic nucleus) have been used to classify pyrillium ring transformations. Different types of mathematical models have been suggested to describe and classify chemical reactions; some of them use ideas of graph theory, for instance, to analyze the redistribution of bonds via chemical reaction [11 - 15]. However, tautomerism of heterocycles (especially typical for substituted heteroaromatics) makes difficult the general use of such formal models (e.g. graphs of bond redistribution) for the general description of heterocyclic rearrangements and ring transformations (see the discussion in ref. [16]). In our early review on ring transformations of azoles, we had
© J.C. Baltzer AG, Scientific Publishing Company


66

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

an idea for a new type of reaction graphs for the needs of heterocyclic chemistry [16,17]. This type of graphs, which neglect tautomerism- the ring-bondredistribution graphs - has been proposed to formalize and classify the known recyclizations of 5-membered heterocycles and to predict some unknown sorts of azole rearrangements. The present article is devoted to the further development of this approach: a brief analysis of some combinatorial properties and relations for the ring-bond-redistribution graphs and an application of the approach to enumeration of degenerated heterocyclic rearrangements.
2. Definitions

We shall use the term heterocyclic ring transformation (HRT) for any reaction containing the steps of heterocyclic ring opening and ring closure in any sequence; the only requirement is that at least one (or more) atoms of the starting heterocyclic ring should be incorporated as a fragment in the final ring. HRT can contain any number of starting reagents and any number of final products. We call the HRT to be simple (SHRT) if (a) only one ring of the starting heterocycle is transformed into only one ring of the final molecule, (b) there is no transient formation of cycles except the formation of the final cycle, and (c) there are no transient permutations of the atoms belonging either to the starting or to the final heterocyclic rings. A lot of known molecular HRTs are SHRTs; this is also true for most of the known heterocyclic rearrangements (HCRs), particularly for such well-investigated "named" reactions as the Dimroth, Comforth, and Boulton-Katritzky HCRs for azoles and azines [1,2,9, 10]; the reactions in figs. 1 and 2 are also SHRTs*. To avoid confusion with the fused heterocycles, let us consider their HRTs to be SHRTs if fused rings have only one pair of common atoms (e.g. as is usual for benzannulation in indole, quinoline, etc.), and if the presence of fused rings does not violate conditions (a)-(c). Let us now consider any SHRT. The atoms of the final structure should be numbered in accordance with the numbers of the starting structure and the mechanism of HRT. Two types of graphs - molecular graph and ring-bond-redistribution graph of SHRT- are determined. A molecular graph (M graph) of SHRT is determined for starting reagents (symbol s) and final products (symbol f) as the pairs Ms(V, Rs) and Mf(V, Rf), where the vertices V and the edges R of the M graph correspond only to the atoms and bonds which are present either in the starting or in the final cycle. The symbols *Reactions (1), (3)-(7) are taken from reviews [1,3], reaction (2) from ref. [19].


E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

67

(1)
+/'7
R2

~"

R'I

~
R2

\N
I

i/

3

~

5

0

8

la
3 3

lb

/
5

2

3
Fig. 1.

of heteroatoms, as well as hydrogen atoms, multiple bonds, substituents and any fused rings should be omitted in the M-graph structure. Examples of M graphs la and lb for SHRT (1) are shown in fig. 1. M graphs are obviously monocyclic graphs; both Ms and Mf graphs contain an equal number of vertices but differ by the number and/or distribution of edges. For SHRTs that are usual intramolecular HCRs, M graphs are connected. Non-connected M graphs correspond to intermolecular SHRT only if the part of the starting (final) cycle is eliminated (entered) via reaction. This pair of molecular Ms and Mf graphs is used to construct the graph of the SHRT-reaction - the ring-bond-redistribution graph (RBR-graph) or GI graph: Ms


68

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

5

7

Z

O~N'*

R

(2)

,

02N

cOR 1

v-

"N"
H
4

"COR 1
4

.4

:,
2

"
'

(3)
NHR
2

3

5C

GI
1 z
?

NHR
"'12

R~

R

5

5

3

-,
3

c;

C=N

(4 )
........ >

0=CH HN N" R

GO ~ ~ 665
(5

R

u

GI
4

GO

(5)
c~

~

,1

4-

C;

+ NH~R 2
~1 "lz

"g

+ R E

1

NH2R

t

.,.-

_'_-,.

f

'~
4

"

665

Ra E

:. . . .
Z ~ Ci

(6)
~J

,
5

E
NMe:: 2

Me

N m

t +

N I',Io

m

E +

664
E--C==-C--E

Me2N-C_=N

.*N

N~
,

OHC~

~

N
IN

a

Z

H
Fig. 2.

563

and Mf graphs of the starting and final structures are "superposed" according to the matching vertices with identical numbers. The resulting G1 graph of reaction contains the same number of vertices (as in M graphs) and edges of different types, designated by solid lines, if the edge is present in both Ms and Mf graphs, dashed lines, if the edge is present only in one M graph, and boldface lines, if the edge belongs to both cycles of M graphs.


E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

69

The necessity of each type of labels in the GI graph is evoked by a reasonable requirement to present maximal information on the appearance (disappearance) or conservation of any skeletal ring bond of the final (starting) cycle at the same structure of a reaction graph. An example of the superposition 2 of M graphs la and lb and the resulting G~ graph 3 for rearrangement (1) are shown in fig. 1. Followig the definition, RBR graphs of direct and reverse reactions coincide. .

General properties and utilization of RBR graphs
The first important property can be formulated as:

THEOREM An RBR graph G~ for any SHRT is bicyclic; its structure corresponds to a pair of cycles with at least one common vertex. The proof of this theorem simply follows our definitions of the simple HRT, M and RBR graphs. In fact, both Ms and Mf graphs contain only one cycle; any vertex and any edge of Ms, Mf or G1 graphs belongs either to the starting cycle or to the final cycle. All these vertices and edges are present in the structure of the RBR graph, so it cannot contain more or less than two cycles. There are also no vertices or edges which do not belong to one of the two cycles, so the structure of the RBR graph can be only the pair of cycles with at least one common vertex. In general, one can find three cycles in the structure of RBR graphs (e.g. the consequences 1234781, 1234561, and 1874561 for the Gl graph 3 in fig. 1); however, only two cycles are linearly independent for the graphs with cyclomatic number 2 [20]. To avoid confusion, we select the cycle to be independent only if it contains boldface-labelled edges, and below use the term cycle of RBR graphs only in this sense. Because each cycle of the bicyclic RBR graph has its ancestor, the cycle of the Ms or Mf graph, the pair of numbered molecular M graphs is in one-to-one correspondence with the G1 graph. It is easy to restore the pair of M graphs from the given graph G1 (operation reverse to superposition in fig. 1): let us choose any of the cycles from the G1 graph; the dashed line(s) of only this cycle should be changed back to solid one(s), while the dashed line(s) of another cycle should be removed. This operation restores one of the M graphs. Now the same procedure should be repeated starting from the second cycle of the G1 graph to restore another M graph. The retaining boldface labels should also be removed. In this manner, the G1 graph gives rise only to the pair of numbered M graphs, but not to the direction of reaction presented by these graphs. Let us define two different types of reaction RBR graphs [16]: the first one, G1 graph; another type of RBR graph, Go graph. It can be constructed from the G1 graph by changing all the dashed-marked edges to solid ones, i.e. it is a bicyclic


70

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

graph with only two sorts of edges (solid and boldface). Corresponding examples of the Go and G1 graphs for some known pyridine SHRTs (2)-(7) are shown in fig. 2. These two constructions of Go and G1 graphs permit us to pose and solve the following series of problems, which seem interesting both for mathematical and heterocyclic chemistry: (1)

general classification and codification of SHRTs, (2) enumeration of the possible types and classes of SHRTs, (3) problem of prediction of the unknown SHRTs.

All these problems are in correlation with each other; some concrete examples of how to solve them for particular cases of azole monocyclic HCRs have been discussed earlier in a review [16]. Let us briefly illustrate the use of our methodology, its generalization, and some important combinatorial results on the examples of pyridine SHRTs.

Types and classes of SHRTs. The construction of the Go graph can be used
to reveal the degree of topological similarity of different SHRTs. Let us say the reactions belong to the same type if their Go graphs coincide, and to the same class when their G1 graphs are identical. As one can see, there are three different types of pyridine SHRTs, namely, eqs. (2)-(5), eqs. (1) and (6), and eq. (7). Inside the type, reactions can differ by class: HCRs (2) (examined earlier by one of us [19]) and (4) are of the same class as the Dimroth rearrangement (3) but differ from the class of SHRT (4); classes of SHRTs (1) and (6) are also different. This clear structural classification of SHRTs is in contrast with traditional (and some vague) utilization of the "named" reactions; for instance, SHRTs (1) and (4), which differ either by Go or by G1 graphs, were earlier related to the same reaction type [3].

Codes for SHRT types. The simple bicyclic structure of Go graphs permits us to define a simple code for any type of SHRT. Let K and L be the size of cycles for a pair of M graphs, and N be the size of their common ring fragment (1 < N < K, K < L). Then the structure of the Go graph can be considered as a pair of annulated K- and L-membered cycles with an N-membered boldface-labelled bridge. The vector (K, L, N) is in one-to-one correspondence with the certain Go graph and can be used to codify the type of SHRT. The codes KLN for Go graphs of reactions (2)-(7) are shown in fig. 2. Enumeration of SHRT classes. Comparing the structures of G1 and Go graphs, one can conclude that the G1 graph is the Go graph with labelled edges; one can easily obtain the possible G1 graphs from an ancestor Go graph by changing some of its edges (designated as solid lines) to dashed ones. Since dashed edges correspond to the formed and broken ring bonds, in the particular case of HCRs only two edges of different cycles of the Go graph should be labelled. (The boldface-marked edges of the bridge should not be labelled, since they correspond to the unchanged ring


E.V. Babaev, N.S. Zefirov, Ring-transformation

graphs

71

bonds in SHRT.) All inequivalent manners of this labelling enumerate the number of different classes of rearrangements in the given type; an example is shown in fig. 3 for the particular Go graph with the code 664. The idea of the described enumeration of G1 graphs is realized in our computer program [18] for any Go graphs with the number of dashed labels from 2 to 4; simple combinatorial equations for enumeration of G1 graphs have also been suggested for rearrangements [16].

3

E = (l.)~.~. (~)(d)(5)(6 ~ (7~(8); Cё = (l) (2) (5;) (4) (57) (68)

664-~
= (i4)(2J)(67)(583

664-~c
2t

~( = (I4)
87 ~"

(2~'~) (56) (78)

664- bo
3

8~6
664-~

~

F:

(E) Fig.3.

Codes for SHRT classes. A simple convention can be used to codify any Gz

graph on the basis of the dashed label distribution. Since the Go graph of the code
KLN contains two annulated cycles, let us choose any node of degree 3 in the Go graph and denote non-boldface edges in both cycles by the letters a, b, c .... (starting

from the a-edges adjacent to the chosen node). As a consequence, an expression
KLN (iKjK. · · ) (iLjL. · · ), showing the position of dashed labels codifies a certain

G1 graph or a certain class of SHRT (the first is the smallest cycle; for cycles of an equal size, the letters should be of lexicographic order). Previously, the idea of such codification was applied to HCRs of azoles [16]; it is now clear how to obtain the corresponding G1 graphs and codify any HCRs. In particular, all the possible codes of pyridine HCRs should have the notation 66N(i)(j) (or simply 66N-ij), see the examples in fig. 3. Moreover, it is easy to codify SHRTs that are not HCRs: the only difference lies in the number of dashed edges


72

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs
of the is not to be (iLjL),

used as labels of the Go graph and, evidently, in the corresponding number letters in the code. In practice, the largest number of formed and broken bonds more than 4, as it is for SHRTs (5) and (6) in fig. 2 (two pairs of ring bonds formed and to be broken). The resulting codes should have the notation KLN-(irjx) as for instance, 665-(ab)(ab) and 664-(ac)(ac) for reactions (5) and (6).

Hierarchic classification of SHRTs. The properties of RBR graphs discussed above permit us to suggest the first classification of simple heterocyclic ring transformation, which is the hierarchic one. In fact, on the first level only the sizes of K and L of the starting and final heterocycles are known. The pair (K, L) determines all the possible vectors (K, L, N): N = 1, 2 ..... K, K < L. These vectors form the second level of classification, level of Go graphs (or of the SHRT types) with the codes KLN. Each Go graph, due to its structure and symmetry, determines a new level, the level of G1 graphs (or of SHRT classes) with the definite codes, e.g. KLN-(i) (j) for HCRs. Each G1 graph determines the corresponding pair of numbered M graphs and can be either directly compared with the real SHRT or used for the deeper levels of classification (involving, for instance, such labels as heteroatoms, substituents, fused rings, etc.). The suggested classification of SHRTs is the full one (for not-simple HRT, see ref. [16]). This means that an expert chemist can easily use this classification tree not only to analyze the similarity of different ring transformations and distributions of known SHRT between the different classes but, more importantly, to predict unknown and chemically reasonable examples of simple HCRs and HRTs. The practical realization of the suggested classification (involving also deeper levels) is our computer program "GREH" (Graphs of REcyclizations of Heterocycles, announced in ref. [18]), which can be used both as a database of known ring transformation and as a tool for prediction of unknown types of HCRs (see examples of some predictions in ref. [16]).

4.

Degenerated ring-transformations of heterocycles

One specific sort of HRTs are degenerated heterocyclic rearrangements (DHCR). Elegant experimental examples of these ring transformations (often proved by the labelled atoms method or by temperature controlled NMR studies) are known for many monocyclic hetarenes, especially for 5- and 6-membered rings [1- 10]. As in the case of any other HRTs, a lot of DHCR are simple. An example of DHCR is the reaction (3) if RI= R 2, see fig. 2. More extensive is the bibliography on the quasi-degenerated rearrangements, different from the DHCR only by substituent presence and/or by tautomerism (e.g. the same reaction (3) if R l differs from R 2 can be considered as quasi-DHCR). An attractive peculiarity of quasi-DHCR is an illusive effect of the substituent(s) migration or exchange and a unique possibility to control the position of substituents (and/ or multiple bonds) only by the temperature or pH variation; on the other hand, the


E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

73

existence of quasi-DHCR is often a reliable signal to search its degenerated prototype. Some reviews on HRTs include DHCR and quasi-DHCR [1-3, 8,9]; nevertheless, it appears that these reactions have never been analyzed from a combinatorial point of view. It seems we are the first to pose the question: how many degenerated rearrangements (which are SHRTs) are theoretically possible for the given heterocyclic nucleus? Although the general graph-theoretical constructions for different types of degenerated rearrangements have been discussed in the literature [21,22] with the following serious group-theoretical analysis of their reaction graphs [23, 24], we did not find an answer to our question. The reason seems clear: before enumeration of DHCR for the given heterocycle, one should define the type (class) of this reaction (or a corresponding reaction graph). Because the general classification of possible types (classes) for SHRTs and HCRs now exists, our approach permits us to solve the enumeration problem of the simple DHCR. Let us consider any SHRT of a K-membered heterocycle, which is DHCR; following our definitions, one can easily construct the corresponding M and RBR graphs. For degenerated SHRTs, the Ms and Mf graphs are obviously isomorphic (the only numeration of vertices in these two graphs is different). Let N be the number of vertices in the ring fragment common to both isomorphic M graphs (1 < N < K, we do not consider rare examples of N = 1); then the corresponding bicyclic Go graph with the code KKN can be defined immediately. Let A, B (0 < A < B) be the number of vertices in two side chains of the M graph (let us call them A and B fragments or chains); then, the K-membered cycle of one M graph should be constructed from these A and B chains and from the N-membered ring fragment, i.e. A+B+N=K. Recall the isomorphism of Ms and Mf graphs; let us carry out the usual superposition (as in fig. 1) of these M graphs to obtain the G1 graph; this includes the correspondence between the matching N-membered fragments of M graphs, while the chains A and B of one M graph should be placed at the (K-N)-membered fragment of the cycle of another M graph. It is possible in a unique manner ifA = B, and there are two inequivalent possibilities to superpose if A < B. The resulting structures of G1 graphs with the different relative disposition of the A and B fragments are symbolically shown as diagrams 4-6 in fig. 4, where the dashed edges of the

4

5

6

Fig. 4.


74

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

G~ graph are located between the terminal vertices of the A and B fragments. Since there are no other variants to superpose an isomorphic M graph, diagrams 4-6 exhaust all types of GI graphs possible for DHCR. Clearly, diagrams 4-6 of fig. 4 present three different types of symmetry of G1 graphs. The symmetry of a G1 graph (as of any other graph) can be characterized by the group of automorphisms Aut(/-'); this group is isomorphic to a certain permutation group determined, for instance, on the vertices of a graph [20]. Since the G1 graph contains a set of labelled edges (solid, dashed and boldface), the only condition for permutations is to conserve both the incidence and the "color" of an edge between a pair of vertices. Diagrams 4-6 present G1 graphs with three different groups of automorphisms; let is call these groups Aut(~) = (e, a), Aut(Fz) = (e, fl) and Aut(F3) = (e, or, [3, ~), where e is identical and a,/3, 7 are non-identical permutations of vertices. One can conclude that in the manner of permutation of the G1 graph vertices (e.g. of the vertices from the bridge or from A and B fragments of diagrams 4- 6), these groups El, F2 and E3 are isomorphic corresponding to the point groups C2v, C2 and D2h. Evidently, the GI graph can be the RBR graph of the degenerated rearrangement only if its structure corresponds to any of diagrams 4-6 in fig. 4, i.e. if its group is of type Aut(Fi), i = 1, 2, 3. Examples of these groups for the concrete G1 graphs are illustrated in fig. 3. (The last G1 graph in this figure with the code 664-(ab) can never be the graph of any DHCR due to its group Aut(/-) = (e): there is no ancestor pair of isomorphic M graphs with such a G~ graph.) Now it is possible to analyze our question of DHCR enumeration for the given heterocyclic ring. Let us consider the simplest case of heterocyclic systems with only one heteroatom, e.g. such heteroaromatics as 5-membered pyrrole, furan and thiophene, or 6-membered such as pyridine, pyrillium or thiapyrillium nucleus. Some known examples of DHCR for these hetarenes can be found elsewhere [1-10]. Any heteroatom can be considered as the label for a vertex of both M graphs, and (after superposition) also for vertices of the G1 graph. Due to only three possible (for DHCR) groups Aut(~) of GI graphs, the following requirements should be used for labelling the vertices of the GI graph: (1) for the vertices vi and v i of the GI graph, which are permuted by permutation zr of the groups Aut(Fi), i = 1, 2, 3, labelling of vi requires labelling of v i, and (2) vi and vi should belong to the different cycles (in the above-mentioned sense) of the G1 graph. (This requirement follows our consideration of a heterocycle with only one heteroatom, i.e. one label in the M graph.) Let us consider a K-membered heterocycle that undergoes all possible DHCR to be enumerated. For the given value N (fixed Go graph), the possible G1 graphs of DHCR are differing by their groups Aut(Fi) and the sizes of the A and B fragments. For the chosen G~ graph, all the inequivalent possibilities of its labelling by one heteroatom (following the above conditions (1) and (2)) correpond to the required number of DHCR and can be calculated in the following manner:


E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

75

(a) for group F~ (diagram 4): A + B + N variants. It is possible to label any pair of vertices that belongs to A (or B) fragments of the G1 graph and are permuted by permutation ct (i.e. A + B variants) and any vertex from the bridge (i.e. N variants); the resulting value is equal to the size K of the heterocycle; (b) for group F2 (diagram 5): A + B + (N mod 2) variants. It is possible to label any pair of vertices from fragments A (or B), which are permuted by permutation fl (i.e. A + B variants), and only central vertex of a bridge if its size is odd (i.e. (N mod 2) variants). Labelling of any other vertex from the odd bridge (or any vertex from the even bridge) violates condition (2); (c) for group F3 (diagram 6): 2A + (N + 1)/2 variants. Because A = B, pair of vertices from fragments A (or B) which are permuted by permutations a fl can be labelled (i.e. 2A variants), and only those vertices from a bridge that non-equivalent to the action of permutations fl and ~ and remain immovable to action of permutation ct (i.e. (N + 1)/2 variants).
any and are the

Now the proposed formulae should be consequently applied for all other Go graphs as ancestors of G~ graphs. Due to the hierarchic classification, the size of cycle K determined possible Go graphs with different N> 1, and the symmetry requirements select appropriate G1 graphs to be labelled with possible A and B fragments. The resulting sum of all labelled G1 graphs should give the number of all possible degenerated SHRTs for the given heterocycles with only one heteroatom. Let us illustrate the use of our formulae for enumeration of DHCR for ring systems of pyrrole, furan and thiophene. There are only three possible Go graphs with N > 1 corresponding to SHRTs of 5-membered heterocycles into othe 5-membered rings; their codes are 552,553, and 554. Among all possible G1 graphs with only a pair of dashed-marked edges (which is necessary for any HCRs) there are only nine graphs, whose groups are F1, F2 or F3 with the following codes: group FI: 552-(a)(a), 552-(b)(b), 553-(a)(a), 554-(a)(a); group F2: 552-(a)(d), 552-(b)(c), 553-(a)(c), 554-(a)(b); group F3 : 553-(b)(b). The size of fragments A, B and N for these Gx graphs and the possible number of labellings is shown in table 1. In the case of DHCR of 6-membered rings with one heteroatom (e.g. pyridine, pyrillium or thiapyrillium), the possible Go graphs have the codes 662, 663, 664 and 665. There are only fourteen GI graphs with groups F1, F2 or F3 whose codes, A, B and N values, and the possible number of rearrangements are also shown in table 1. These data exhaust all the number of DHCR (evidently, which are SHRTs) for 5- and 6-membered heterocycles with only one heteroatom. As an example, all


76

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

1
8

:~
2

N

z I/"~N

e

8

6

c~

N/
1 tt

3

3

I

z

4 N?
~- .....

N
c5 8

~
c~

l

3

U

C)

8

> c"

l

3

8

C5

B 0 1 ~3

,c'i

G 6
1

3

U

2

~

C~

1

~3

Fig. 5.


E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

77

Table 1 5-membered rings group and code N A B of GO graph F1: 552-aa 552-bb 553-aa 554-aa 2 2 3 4 03 12 02 01 number of DHCR 5 5 5 5 6-membered rings code of G l graph 662-aa 662-bb 663-aa 663-bb 664-aa 665-aa 662-ae 662-bd 663-ad 663-bc 664-ac 665-ab 662-cc 664-bb NAB 2 2 3 3 4 5 2 2 3 3 4 5 2 4 0 1 0 1 0 0 0 3 0 1 0 0 4 3 3 2 2 1 4 1 3 2 2 1 number of DHCR 6 6 6 6 6 6 4 4 4 4 2 2 5 4

F2: 552-ad 552-bc 553-ac 554-ab

2 2 3 4

03 12 02 01

3 3 3 1

F3: 553-bb

3

1

1

4

22 11

the labelled G1 graphs with the code 664-(a)(a) and the corresponding labelled pairs of M graphs are shown in fig. 5 for the sekeleton of the pyridine nucleus. We would like to mention that the only known example of degenerated SHRT of pyridine seems to be the above discussed Dimroth rearrangement (3) in fig. 2. The results would also be applied in the search for new quasi-DHCR.

References [1] H.C. Van der Plas, Ring Transformation of Heterocycles, Vols. 1, 2 (Academic Press, New York, 1973). [2] G. L'abbe, J. Heteroc. Chem. 21(1984)627. [3] A.N. Kost, S.P. Gromov and R.S. Sagitullin, Tetrahedron 37(1981)3423. [4] O.P. Shwaika and V.N. Artyomov, Usp. Khimi 41(1972)1788. [5] A.T. Balaban et al. (eds.), Pyrillium Salts, Syntheses, Reactions and Physics Properties (Academic Press, New York, 1982). [6] A.T. Balaban, in: New Trends in Heterocyclic Chemistry, ed. R.B. Mitra et al. (Elsevier, Amsterdam, 1979), p. 79. [7] A.T. Balaban, in: Reports of Vl Int. IUPAC Congress on Organ& Syntheses, Moscow (1986). [8] A. Padwa, in: Rearrangements in Ground and Excited states, Vol. 3, ed. P. de Mayo (Academic Press, New York, 1980), p. 501. [9] A.J. Boultort, in: Lectures in Heterocyclic Chemistry, Vol. 2, ed. R.N. Castle and L.B. Townsend (Heterocorporation, Orem, 1974), p. S-45; M. Ruccia, N. Vivona and D. Spinelli, Adv. Heteroc. Chem. 29(1981)141. [10] A.R. Katritzky and C.W. Rees (eds.), Comprehensive Heterocyclic Chemistry (Pergamon, Oxford, 1984).


78

E.V. Babaev, N.S. Zefirov, Ring-transformation graphs

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