The number of Kekule structures of all conjugated systems
containing a linear polyacene chain of length n is a linear function
of the parameter n and is given by eq. (8) 

Ivan Gutman

English

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CROATICA CHEMICA ACTA CCACAA 55 (4) 371-374 (1982) CCA-1341 YU ISSN 0011-1643 UDC 547.52 Original Scientific Paper The Number of Kekule Structures in Conjugated Systems Containing a Linear Polyacene Fragment* Ivan Gutman Faculty of Science, University of Kragujevac, P. 0. Box 60, 34000 Kragujevac, Yugoslavia Received February 8, rns2 The number of Kekule structures of all conjugated systems containing a linear polyacene chain of length n is a linear function of the parameter n and is given by eq. (8) . In a recent paper1 Eilfeld and Schmidt reported closed formulae for the number of Kekule structures for a variety of homologous series of ben-zenoid hydrocarbons containing a linear polyacene fragment. All their for-mulae were of the form K 0 =a+ b n (1) with a ood b being constants, depending on the homologous series considered. In the present paper we demonstrate that formulae of the form (1) hold for all conjugated molecules (not necessarily benzenoid), containing a linear polyacene fragment. In addition, we will clarify the topological nature of the constants a and b. We will use standard graph-theoretical notation and terminology2• The molecular graph of a conjugated system containing a linear polyacene fragment of the length n is represented by L 11 • p t GB ~--·~ u "' Ln Lo 8: s'© ~$ q · · · n-1 n 5 r: A B Mn P2 * Part XIII of the series »Topological Properties of Benzenoid Systems« 372 I. GUTMAN The subgraphs A and B symbolize certain (but arbitr ary) terminal groups. Hence Ln can be understood as being obtained by annela1Jion of the fragments A and B to both ends of an n-cyclic linear polyacene molecule. The positions where the annelations occur will be labelled by p, q and r, s, respectively. In the special case when the subgraph B is isomorphic to the graph P 7 (possessililg just two vertices r and s), instead ·of Ln we have the molecular grnph Mn. The molecular graph of the n-cyclic linear polyacene is of course, also a special case of Mn and Ln. The number of Kekule structures of a conjugated system, whose molecular graph is G will be denoted by K (G). Let V r and V s be two adjacent vertices of G and let the edge between them be denoted by ers· Then according to a well known result3, K (G) conforms to the following recurrence r elation K(G) = K(G - ers) + K(G- vr- vs) (2) If the vertex V r is of degree one (i. e. if Vs is the unique neighbour of Vr) , then from (2) we obtain K (G) = K (G-vr-vs> (3) Hence, the deletion of a vertex of degree one and of its first neighbour will not ·alter the number of Kekule structures. We determine first the number of Kekule structures of the system M n. Applying eq. (2) to the vertices V r and V s of M n we get K (Mn) = K (Xn_1) + K (Y n- 1), where the graphs X n-l and Y0 _ 1 are given as follows. Yn-1 Both X0 _ 1 and Y n- l possess vertices of degree one. Then by repeated appli-cation of (3) we easily conclude that K (Xn-1) = K (Mn-1), K (Yn-l) = K (A-vp-vq). Hence we deduced the recurrence relation K (Mn) = K (Mn_1) + K (A - VP - vq), from which one immediately finds the general solution K (Mn)= K (A)+ n K (A-vp - v q), since for n = 0, Mn coincides with A. (4) (5) (6) TOPOLOGICAL PROPERTIES OF BENZENOID SYSTEMS 373 Using the auxiliary result (6) we can now calculate the number of Kekule structures of the system Ln. The application of the identity (2) first to the edge ert aind then to the edge esu of Ln gives K (Ln) = K (Yn_1) K (B) + K (Yn_1-vt) K (B-vr) + K (Yn-l -vu) K (B-v,) + + K (Mn_1) K (B - vr - vs) . The graphs Yn-1 - Vt and Y 11_ 1 - Vu possess vertices of degree one and eq. (3) can be repeatedly applied. Finally we get K (Yn_1 -vt) = K (A-vq) and K (Yn_1 -vu) = K (A-vP), which together with (4) gives Therefore, K (Ln) = K (A - vp-vq) K (B) + K (A- vP) K (B - v s) + + K (A-vq) K (B-vr) + K (M11_ 1) K (B-vr-vs). K(Ln) - K (Ln_1) = [K (M11_ 1)- K (Mn_2)] K (B-vr- v.) and because of (5), K (Ln) = K (Ln_1) + K (A-vp - Vq) K (B - vr-vs). The general solution of the recurrence relation (7) is K (Ln) = K (L0 ) + nK (A-vp-vq) K (B-vr-vs)· (7) (8) Note that the above equation is a generalization of (5) . In fact, for B = P 2, eq. (5) follows from (8) because of K (B - Vr - V s) = 1 and Lo = A. Hence we proved that the formula (1) is the general expression for the number of Kekule structures of all Hnear polyacene derivatives, irrespective of the nature of the terminal groups. In practice it often happen that we know the number of Kekule structures of Ln for certain values of n . If we know K (Ln) for two fixed values of n, say for n = g and n = f, then we can use the formula f K (Lg) - g K (Lf) + [K (Lr) - K (Lg)J n K(Ln) = ---=-----------~ f-g In particular, in the most usual case when g = 0 and f = 1, we have K (Ln) = K (L0 ) + [K (L1) - K (L0 )] n . or or Comparing the formulae (8)- (10) with eq. (1) we see that a=K(L0 ) and b =K(A-vp - vq)K(B-vr-vs) a= f K (Lg) - g K (Lf) f-g and b = _K_(_L_r)_-_K_(L-'g~) f-g a = K (L0 ) and b = K (L1) - K (L0 ). Of course, (13) is a special case of (12) . (9) (10) (11) (12) (13) 374 I. GUTMAN The equations (11) provide an elementary route for the construction of closed expressions for the number of Kekule structures of any particular homologous series. As an example, consider the tetrabenzacenes, T n · ~--·~ Tn Zn Here A and B are isomorphic with the phenanthrene graph, A - Vp - Vq and B-vr - Vs are isomorphic with the biphenyl graph and T 0 is the molecular graph of 1,2,7,8-dibenzochrysene. Since K (To) = 24 and K (A -- Vp - Vq) = K (B - Vr - Vs) = 4, we immediately obtain the formula K (T n) = = 24 + 16 n, which is in full agreement with ref. 1. We would also like to point out the following special case. If any two members of a homologous series have an equal number of K ekule structures, then by eq. (12) , b = 0 and therefore all members of this homologous series must have an equal number of Kekule structures. Similarly, if either A - Vp - Vq or B - Vr - Vs (or both) have no Kekule structures, then the recurrence relation (7) becomes K (Ln) = K (Ln_1) and thus the number of Kekule structures of Ln is independent of the length of the polyacene chain. This always happens when the terminal groups A and B have an odd number of vertices. A typical example is the zethrene series (Zn), all members of which have nine Kekule structures. REFERENCES 1. P. E i 1fe1 d and W. Schmidt, J. Electron Spect r. Rel. Phenom. 24 (1981) 101. 2. See for example: A. Gr a ova c, I. Gutman, and N. Trina j st i c, Topolo-gi cal Approach to the Chemistry of Conjugated MoLecules, Springer-Verlag, Ber-lin 1977. 3. M. Randi c, J. Chem. Soc. Faraday II, 72 (1976) 232. SAZETAK Broj Kekuleovih struktura konjugiranih sustava koji sadrze linearni poliacenski fragment Ivan Gutman Za sve konjugirane sustave koji sadrze linearni poliacenski lanac duljine n, broj Kekuleovih struktura je linearna funkcija parametra n i odreden je jed-nadzbom (8). 

The Number of Kekule Structures in Conjugated Systems Containing a Linear Polyacene Fragment

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