J. Chem. Znf. Comput. Sci. 1994, 34, 357-360
357
Calculating the Cell Polynomial of Catacondensed Polycyclic Hydrocarbons Peter E. John University of Technology of Ilmenau, D-98693 Ilmenau, Germany Received May 18, 1993" In the paper we establish a simple algorithm of low complexity for calculating the so-called cell polynomials for a catacondensed polycyclic unsaturated hydrocarbon H, which enables the KekulC structure count (KSC) and the algebraic structure count (ASC) of H to be calculated. 1. INTRODUCTION
During the past 2 decades applications of graph theory to polycyclic aromatic hydrocarbons have raised great interest in resonance theory. In this respect, Herndon's resonance theory1P2and the conjugated circuit model introduced by RandiC3*4are of considerable importance. The sextet polynomial found by Hosoya and Yamaguchis allows a systematic combinatorial enumeration of KekulC structures of (catacondensed) aromatic hydrocarbons. This polynomial was shown to possess a number of interesting properties and to reflect Clar's resonant sextet theorya6 Various further developments of the sextet polynomial concept can be found, e.g., in refs 7-9. In the present paper we establish a simple algorithm for calculating the so-called cell polynomials for a catacondensed polycyclic unsaturated hydrocarbon H, which enable the Kekult structure count (KSC) and the algebraic structure count (ASC) of H to be calcuated. In connection with the ASC of H see, e.g., the papers by Wilcox,lo Herndon," Klein et al.,12 Dias,13 and refs 14 and 15. 2. DEFINITIONS AND NOTATION
Let e denote the Euclidean plane and let G be the set of all tweconnected finite planar graphs (without loops and multiple edges). Let G = (V,E) E G with vertex set V = V(G) and edge set E = E(G), and let n = n(G) = IVl, m = m(G) = IEl denote the numbers of vertices and edges of G, respectively. Graph G, is an embedding of G into e. G, subdivides e in m - n + 1 =: c = c(G) finite open domains Di, i = 1, 2, ,.., c, the infinite open domain Do and the union of the boundaries of all these domains (Figure 1). The boundaries of the Di are denoted by B, = B(Di), i = 1,2, ..., c; Bo is also called the contour (or periphery) of G,. G, is called a map denoted by M = M(G). Cell Ct is the union Di U Bi, i = 1, 2, ..., c. Note that B(CJ =: Bi. The inner dual D = D(M) of M is the dual of M without the vertex that corresponds to the infinite domain. Map Mis a contour map if D(M) is a tree (Figure 1). A perfect matching (PM) or linear factor of a graph G is a set of pairwise disjoint edges that cover all vertices of G (Figure 5 ) . Case c(M) = 0. Then M = 0 is the zero map (without edges and vertices). By convention, map 0 has precisely one PM. Case c(M) = 1. Then M has exactly one cell which is called an isolated cell. Case c(M) > 1. Edge e E E ( M ) is called a contour edge or an internal edge of Mif the whole edge, or only its endpoints, Abstract published in Aduance ACS Absstracsts, February 15, 1994.
Figure 1.
lieon Bo(Figure 1). Cell CisanendcellofMifCcorresponds to an end vertex of D(M) (in particular, an isolated cell is an end cell). Let denote the set of all contour maps. Map M E )jis bipartite if and only if V = V(M)can be divided into two disjoint subsets P = V(M) and f' = k(M)(V = t U f', P ( kl = 0) such that vertices from the same subset are never adjacent. Let gbdenote the set of all bipartite contourmaps. Note that for every M E Mbfi = fi and M has a PM. Obseruation 1. Every M E Mbwith c ( M ) > 1 has at least two end cells. Obseruation 2. Every M E Mbis the last of a (finite) sequence of bipartite contour maps (Mi),i = 1,2, ...,c, where M I is an isolated cell and for i = 2, 3, ..., c we obtain Mi by adding cell Ci (Ci is an end cell of Mi) to Mi-1 (see Figure 2). Any plane image (Le., an embedding in e) of a (bipartite) planar graph G is called a (bipartite) pattern (of G). The cells of a pattern are defined in an analogous way as above. Let &b denote the set of all (bipartite) subpatterns of all maps from M ~ . 3. DEFINITION OF THE CELL POLYNOMIAL OF A
BIPARTITE PATTERN
ab
For A E let C = C(&) = (Ci1,C,, ..., CiJ and N = N ( A ) = (il, i2, ...,ic)denote the set of all cells and of all cell indices (labels), respectively (note that if C = 0, then N = 0).
QQ95-2338/94/1634-Q35l$Q4.5Q/Q 0 1994 American Chemical Society
JOHN
358 J. Chem. In& Comput. Sci., Vol. 34, No. 2, 1994
M’
A
Figure 2.
M‘ = M o- V(C!) Figure 3.
N there corresponds a cell set G = E I) of M ;I is called an index set of M.
To every subset I
{C& E C and i
The pattern MI is obtained from M by deleting in A all vertices of all cells from G and all edges which are incident with these vertices. Index set IC N is feasible (in A) if (i) any two cells C’, C” E are disjoint (B(C’) n B(C”) = 0) and (ii) MI has a PM. Note that A0 = M ;this implies that I = 0 is feasible if and only if M has a PM. To every cell Ct E c ( M ) is assigned a weight wi = w(Ci) which is an element of an (algebraic) ring (here it suffices to assume that wi is a real number). Cell set of M has weight
s
otherwise = 1 if 0 is feasbile). The cell (this includes WE+) polynomial f d = fh(w1, w2, ..., w,) of M is defined as
For example, consider the pattern M * =: MO - V ( c ) ,given in Figure 3. Here N(M*) = {1,3,4];the only feasible sets are 0 and 11); thus fh*= 1 + ~ 1 . Note that for any bipartite pattern Jn the cell polynomial fm can be defined as described above. 4. CALCULATION OF THE CELL POLYNOMIAL OF
A BIPARTITE PATTERN WHICH IS PART OF A BIPARTITE CONTOUR MAP
Let M E Abwith cell set C, where cell Ci E C has weight wi = w(Ct). The cell polynomialfA can be calculatedapplying
the following recursive procedure. Algorithm f: Cf.0) If M consists of a single (isolated) vertex, put f A = 0; Vl>f0 = 1;
e @ Figure 4.
cf.2) If A has components AI,A2, ..., Ap, put fA = fASAy.fAlP; (f.3) If Jt E &, A is connected,and c(A)> 1, consider the following cases. (i) M has no cut vertex ( A E Mb). Then (because of observation 2) h can be obtained from A, E Bbby adding cell C to Ac(C is an end cell of Jn). Delete in M all vertices of V(C) (and all edges which are incident with these vertices). Denote the resulting pattern by hc.Put fm = fme + W(C)fA
(ii) A has a cut vertex ( A E Mb). (ii.1) M has a hanging edge (u, u) E E(&). Delete vertices u,u and all edges which are incident with them. Denote the resulting pattern by M!. Put
=fn1 (ii.2) A has no hanging edge. Then there exists an induced subpattern AIcof “ A such that Jn” E Mb, A” contains exactly one cut vertex of A, and &It is contained in no other submap of Jn with these properties. Delete in Jn all vertices of M“ (and all edges incident with them); this results in a pattern which we call A’”. Put fA
f A = fAfJAl#t By successively applying f.CLf.3, eventually the polynomial fA of M E Abis found. For example consider the map Mo E Mbof Figure 2. Mohas five cells with weights wp = w ( c > ,i = 1, 2, 3, 4, 5 . Here we find (see Figure 4)
THECELL POLYNOMIAL OF POLYCYCLIC HYDROCARBONS
A
Figure 6.
Figure 5. fhf0
=fM,O =fM,O
+ w;f&* =
***
=fM,O
+ w!fM,O
fM,O
+ wp+l
fD
Therefore,
=i
+ + + + + + W:
W!
W;
W:
W!
q@,Lll) is even (odd) if and only if Ll and are in the same classTin different classes) of the partition. Note that only the partition is determined (not the individual classes); the classes are interchangeable. Put d = d(G) = ILJI and 111 = lll(G) = LIII. Proposition: For every M E M b
-
V*m(l, -111 = I~"(M)- &&)I This proposition can be proved in a way analogous to the proof of Theorem 6 in ref 15. As an example we use the bipartite contour map Mo of Figure 1
and for i = 3, 2, 1, 0 by use of (f.3(ii))
fMo
J. Chem. Inf. Comput. Sci., Vol. 34, No. 2, 1994 359
W:W!
pM(l) = I(M)= 7
It can easily by checked that this result is in accordance with the definition of the cell polynomial (section 3). 5 . COARSENED CELL POLYNOMIALS
The cell polynomial fk =f&wl, w2, ...,w,) of M E Abcan be "coarsened". Let ni = n(Ci) denote the number of vertices of Bi = B( Cj). Inserting wi
=
h:
if ni = 2, mod 4 if ni 0, mod 4
into fm, we find the first coarsened cell polynomial f " =~ f*n(x,v).With x = y =: z we obtain the second coarsened cellpolynomialf**m=f**n(z) =: f*&,z). As an example we use the contour map Mo of Figure 2:
+
= 1 2x+ 3y+
Indeed, by inspection we easily obtain
j ( M 0 ) = 4 , rll(M0) = 3
6. CONCLUDING REMARK
It is worth mentioning that algorithm f can, in a suitably extended form, be applied to any bipartite pattern. For example, consider M 1and Mz given in Figure 6; we have
$*MI
f"*n= K M ) This formula follows immediately from the way the algorithm is constructed; it is well-known for M being a catacondensed hexagonal system.5 A perfect basicfigure (PBF) of graph G is a subgraph U of G with the following properities: (i) Every component of U is a circuit or a dumbbell 0-0; (ii) U covers all vertices of G (Figure 5). Clearly, the union of any two PMs L', L*I E L(G of G is the edge set of a PBF; conversely, every PBF Ucan be obtained this way. We shall denote the PBF Udetermined by the pair [L', L*I] briefly by (L', L*I). Let q(U) = q(L*,L'I) denote the number of circuits contained in U = (L', L'I) whose length is a multiple of 4. Lemma: There is a unique partition LI,LII of _L (i.e.,_L = LI U QI, Q n L_ll = 0 where Ll or may be empty) such that
+ 4x + y + 2x2 =f**Ml(z) = 1 + 5z + 2z2
y M =f*,(x,y)
XY
If there is no danger of confusion, we shall directly transfer concepts and symbols, originally defined for planar graphs, to their plane images (embeddings, patterns). Let L = L ( G ) and 1 = l(G) denote the set and the number of all PMs of G, respectively. Observation 3: For every M E Mb -
(or i(M0)= 3, lIl(M0) = 4)
=1
The significance of this extension will be discussed in a subsequent paper. REFERENCES AND NOTES ( 1 ) Herndon, W. C. Resonance energy of aromatic hydrocarbons. Quantitative test of resonance theory. J. Am. Chem. SOC.1973, 95, 2404. (2) Herndon, W. C.; Ellzey, M. L. Resonance Theory. V. Resonance energies of benzenoid and nonbenzenoidr systems. J . Am. Chem.SOC. 1974,96, 6631. (3) RandiC, M. Conjugated circuits and resonance energy of benzenoid hydrocarbons. Chem. Phys. Lett. 1976,38,68. (4) RandiC, M. A graph theoretical approachto conjugationand resonance energies of hydrocarbons. Tetrahedron 1977, 33, 1906. (5) Hosoya, H.; Yamaguchi, T. Sextet polynomial. A new enumeration and proof technique for the resonance theory applied to the aromatic hydrocarbons. Tetrahedron Lett. 1975,4659. (6) Clar, E. The Aromatic Sextet; Wiley: London, 1972. (7) Aihara, J. Aromatic sextets and aromaticityin benzenoidhydrocarbons. Bull. Chem. Soc. Jpn. 1977, 50, 2010. (8) Gutman, I. A method for calculation of resonance energy of benzenoid hydrocarbons. Z . Naturforsch. 1978, 33a, 840. (9) Gutman, I.; Cyvin, S. J. Introduction to the Theory of Benzenoid Hydrocarbons; Springer: Berlin, 1989. (10) Wilcox, C. F. Stability of moleculescontaining(4n)-rings. Tetrahedron Lett. 1968, 795. (1 1 ) Herndon, W. C. Enumeration of resonance structures. Tetrahedron 1973, 29, 3.
360 J. Chem. In5 Comput. Sci., Vol. 34, No. 2, 1994 (12) Klein, D. J.; Schmalz, T. G.; El-Basil, S.; RandiC, M.; Trinajstib, N . KekulC count and algebraic structure count for unbranched alternant catafusenes. J. Mol. Srrucr. (THEOCHEM)1988, 179,99. (13) Dias, J. R. Note on algebraic structure count. 2.Narurforsch. 1989, 44A, 761.
JOHN (14) John,P. E. Commentonthepaper: KekulCcountandalgebraicstructure count for unbranched alternant cata-fuscnes ( J . Mol. Srrucr.
(THEOCHEM)1988,179,99). (15) John, P. E. Lmearfaktoren indefekten hexagonalen Systemen. Bayreurh-
er Math. Schrifren, 1993,43,35.
