The previous work by Bonchev et al.3 is complemented by an
explicit general topological formula for Hosoya\u27s index of a cycle

I. Gutman

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CROATICA CHEMICA ACTA CCACAA 54 (1) 81-83 (1981) CCA-1260 YU ISSN 0011-1643 UDC 539.19 Note On the Characterization of Monocyclic Structures. Hosoya's Index I . Gutman* Institut fiir StrahLenchemie im Max-PLanck-Institut filr KohLenforschung, Mii!heim a. d. Ruhr, F. R. Germany Received May 15, 1980 The previous work by Bonchev et al. 3 is complemented by an explicit general topological formula for Hosoya's index of a cycle. A large number of topological indices which characterize various topo-logical properties of molecules have been proposed in the current chemical literature. (For a recent review see1.) The main chemical applications of these indices are in bond-additive and other empirical schemes for correlating molecular properties (e.g. boiling point, density, refractive index, surface tension, viscosity, octane number, chromatographic constants, thermodynamic constants, pharmacological activity etc.) with molecular structure1•2• In the majority of the papers dealing with t.opological indices only acyclic structures were considered. However, in recent investigations by Bonchev et al.3•4 an attempt was made towards the application of topological indices · for the characterization of cyclic structures. Monocyclic systems were examined in3 and polycyclic systems in4. In particular, Bonchev et al. considered3 the following topological indices of the cycle CN: the Wiener number W , Randie's connectivity index XR, Flatt's index F, Gordon's and Scantelbury's index S, the indices M1 and M2 , Hosoya's index Z and the information index Iv. They presented explicit analytical expressions for the indices W , xn, F, S, M1 and M2 , but not the analogous formula for Hosoy a's index. In this note we shall solve the problems posed in ref. 3 for the case of Hosoya's index. Let G be a graph with N vertices and let N = 2 m or N = 2 m + 1. If we denote the number of k-matchings of this graph by p (G, k), k = 1, 2, ... , then Hosoya's index of G is defined as3•5 111 Z (G) = ~ p (G, k) k = O while the matching polynomial of G is given by6 '" a (G) = a (G, X) = ~ (- 1)" p (G, k) x N-z• k =O It is immediately seen that Z (G) = i -N a (G, i) (1) * Alexander von Humboldt fellow. Permanent address: Faculty of Science. University of Kragujcvac, P. 0 . Box GO, 34000 Kragujevac, Yugoslavia 82 I. GUTMAN Let CN denotes the cycle with N vertices. It is well known7 that N ( N -k) p(CN, k)= --N-k k Therefore m N ( N -k) Z(C,v) = ~ N-k k k= O is the required topological formula for Hosoya's index of CN. Another, more convenient expression for Z (CN) is obtained from the relation8 a (CN) = X a (CN_1 ) - a (CN_ " ) which together with (1) yields Z (CN) = Z (CN_1 ) + Z (CN_) The solution of the recurrence relation (2) reads9 Z (CN) = [(v5.f. 1)/2)N + (- l)N [(\I 5-1)/2)N where we have used the initial conditions Z (C3 ) = 4 and Z (C4) = 7. It is now easy to calculate that ti Z (odd_.,.. even) - ti Z (even_.,.. odd) = = - c-15._ 2) [( v5 + 1)/21N + <vs·+ 2) [< v5 - 1)/2t (2) (3) (4) where N is the size of the even-membered cycle. The expression on the right side of eq. (4) has negative values for all N ~ 4 and thus the application of Hosoya's index leads to conclusions about the change in relative cyclicity which are just the opposite to those formulated in Rule 1 in ref.3. In ref.3 it was found that Hosoya's index cannot be normalized by dividing it by the number of carbon atoms. This conclusion is now evident from eq. (3) since we see that Z (CN) increases exponentially with increasing N. We would therefore propose another normalization, namely 1 Z (CN) = N log Z (CN) (5) which has the advantageous property (6) . ti Z (odd_.,.. even) - ti'z (even_.,.. odd) > O (6) Thus Rule 1 in ref.3 can also be justified on the basis of the normalized Hosoya's index, eq. (5). Concluding this note we would like to point out some relations between Hosoya's numbers of a cycle and a path (chain). Let PN be the path with N vertices. It is long known5 that Z (PN) = Z (PN_1) + Z (PN_) and z <P:vl = ~ ( N - k ) = [(v5+ 1)/2r+11..f5 + c-1)N [<-15- 1)/21N•1N5 k = O k CHARACTERIZATION OF M O N OCYCLIC STRUCTURES 83 From the following identities between the matching polynomials of paths and cycles10 w e deduce Consequently, and a (CN) =a (PN)- a (PN_) a (C,v) a (P,v_1 ) = a (P"N_1 ) Z (chain) - Z (cycle) = - Z (PN_2) < 0 - - 1 Z (chain)-Z (cycle)= -- N log [1 + Z (PN_)/Z (PN)] < 0 Acknowledgement. - The author thanks the Alexander von Humboldt Foun-dation for financial support of this research. REFERENCES 1. A T. Ba 1 ab an, Theor. Chim. Acta 53 (1979) 355. 2. L. Ki hr and L. H. Ha 11, Molecular Connectivi ty in Chemistry and Drug Research, Academic Press, New York 1976. 3. D. Bon ch ev, 0. Me ken ya n , J . V. Knop, and N. T r ina j s t i c, Croat. Chem. Acta 52 (1979) 361. 4. D. B on ch e v, 0. Me ken y a n , and N. T r in a j s t i c, Internat. J . Quantum Chem. 17 (1980) 845. 5. H. Hos o ya, Bull. Chem. Soc. Japan 44 (1971) 2332. 6. See for example : I. Gutman, Math. Chem. (Millheim) 6 (1970) 75; C. D. G o d s i 1 and I. Gutman, J . Graph Theory, in press. 7. D. M. Cvetkovic, M. Do ob, and H. Sachs, Spectra of Graphs - Theoru and Application, Deutscher Verlag der Wissenschaften, Berlin 1980, p. 73. 8. I. Gutman and H. Hos o ya, Theor. Chim. Acta 48 (1978) 279. 9. D. C v e t k o vi c and I . Gut m an, Croat. Chem. Acla 46 (1974) 15. 10. I. Gutman, Z. Naturforsch. 35a (1980) 453. SAZETAK O karakterizaciji monociklickih struktura. Hosoyin indeks I. Gutman Kao dopuna rada Bonceva i suradnika3 izvedena je eksplicitna opca topoloska formula za Hosoyin indeks monociklickih struktura. I NSTITUT ZA KEMIJU ZRACENJA MULHEIM, SR NJEMACKA Prispjelo 15. svibnja 1980 

On the Characterization of Monocyclic Structures. Hosoya\u27s Index

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