In recent years conjugated molecules have been intensely studied by
means of graph theory1 and a number of their chemical and physical properties could be described and/or predicted. The aim of the present note is to analyze a special class of molecular graphs2 which are of doubtless importance for chemistry since they represent benzenoid hydrocarbons

Ivan Gutman

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CROATICA CHEMICA ACTA CCACAA 46 (3) 209-215 (1974) CCA-837 YU ISSN 0011-1643 539.19:516 Note Some Topological Properties of Benzenoid Systems Ivan Gutman Institute »Ruder Boskovic«, 41000 Zagreb, Croatia, Yugoslavia Receive d May 27, 1974 INTRODUCTION In recent years conjugated molecules have been intensely studied by means of graph theory1 and a number of their chemical and physical pro-perties could be described and/or predicted. The aim of the present note is to analyze a special class of molecular graphs2 which are of doubtless im-portance for chemistry since they represent benzenoid hydrocarbons. In fact, there is no uniquely adopted agreement as to which conjugated sy;stems are benzenoid, but it seems that most organic chemists call benzenoid those molecules which are burilt entirely from (regular) hexagons. The network obtained by any combination of regular hexagons in a plane such that two hexagons have either exac.tly one common edge or are disjoint will be called a »benzenoid graph«. It is assumed that the considered graphs are connected2, that is, have only one component. For example, I-III are the only possible benzenoid gra.phs with three rings. Of course, they represent anthracene, phe-nanthrene and phenalenyl. respectively. II ill In graph-theoretical literature only the problem of enumeration of ben-zenoid graphs (i.e. the problem how many different arrangements of a given number of hexagons is possible) is studied4 • The nomenclature of benzenoid graphs is discussed in Ref. 5. For theoretical studies of conjugated molecules the graph spectrum2 i:s of utmost importance3• However, the spectral properties of benzenoid graphs seem to be almost completely unknown. In fact, only one exact result is known, namely the relation 6' 7 N II xi = (- l)N/2 K 2 j = l (1) where the set of numbers { x1 , x 2 , ••• x N} irs the graph spectrum, N is the number ·of vertices, and K is the number of Kekule structures. Of course, all 210 I. GUTMAN relations which hold for bipartite graphs2 hold for benzenoid graphs as well. Recently, Hall has analyzed8 these relations and was able to derive a set of approximate formulae for resonance energy, bond orders etc. of benzenoid systems. Additional approximate formulae of this kind can be found in Ref. 9. SOME ELEMENTARY PROPERTIES OF BENZENOID GRAPHS The number of rings (R) is one of the most obvious structural details of benzenoid g:raphs. Since the number of vertices (N) and edges (M) is not uni-quely determined by R, as it can be seen when graphs I--III are compared, the notion of internal ver tex is introduced. A benzenoid graph divides the plane in one infinite and R finite regions (i. e. rings). All vertices and edges which lie on the boundary of the infinite region form the perimeter of the graph. The vertices which do not belong to the perimeter are internal. Their number is Ni. It is easily seen that all internal vertices are of degree2 three, that is to say that they have three adjacent vertices. Therefore the internal vertices belong to three rings. Every internal vertex lies in the interior of a 12-membered cycle. N; can be any non-negative integer. In example IV the internal vertices are labelled by e. · If Ni = 0, the corresponding molecules are oalled5 »Cata-condensed«. Hence, the following statements are equivalent: - Ni=O, - no vertex belongs to three rings, - no three rings are mutually adjacent, all vertices belong to the perimeter, all cycles contained in the graph are of the length 4m + 2 (m = = natural number)7, the dual graph of the molecular grnph contains no triangles10, and any of them can be used as a definition of cata-condensed systems. The following equation N +Ni= 4R + 2 (2) can be simply proved by induction. For F = 1, 2, 3 eq. (2) is fulfilled. Now, let V have R rings and eq. (2) becomes: N (V) + Ni (V) = 4 R + 2 (3a) TOPOLOGICAL PROPERTIES 211 y Let there be A vertices on the perimeter of V between the vertices 1 and 2. If one generates a new ring in VI by joining a chain of B vertices to 1 and 2, it is obvious that But since VI is benzenoid, Ni (VI)= Ni (V) +A N (VI) = N (V) + B A+B+2=6 Substituting eqs. (3b)-(3d) back into (3a) one obtains N (VI) + Ni (VI) = 4 (R + 1) + 2 which completes the proof of eq. (2). Similarly it can be show n that M +Ni= 5R + 1 (3b) (3c) (3d) (3e) (4) Note that eqs. (2) and (4) are a generali?iation of the Euler formula M - N = = R-1. By inspection of graphs V and VI, the equation T = 2 (R-1) (5) follows immediately, where T is the number of vertices of degree three. Actually, the vertices 1 and 2 are of degree two in V and of degree three in VI, while the degree of other vertices is not changed. If S is the number of vertices of degree two and D" the degree of the vertex p, S = 2R-Ni + 4 N }; D~ = 26R-4Ni-2 p = l (6) (7) Because of eq. (1) Kekule structures play an important role in the graph spectral studies of conjugated systems6•7• Hall has demonstrated8 that the total n-electron energy in a series of isomeric benzenoid hydrocarbons is a linear function of K. It can be shown that if a benzenoid graph has at least one Kekule structure, K;;?: 4 (except for R = 1 and R = 2). However, it is not simple to find in the general case the conditions for K > 0. Benzenoid graphs are bipartite, that is their vertices can be separated in a unique manne,r into two groups - »starred« and »unstarred« - such that starred vertices are adjacent to the unstarred and vice versa. Let X and Y denote the number of starred and unstarred vertices, respectively and Z = = !X-Y !. 212 I. GUTMAN Z can be any non-negative integer. This statement, which is not neces-sarily obvious, can be illustrated with the triangulene series VII-X. 0 00 -W YITI IX x z=O Z= 1 Z=2 z=3 In every Kekule structure a starred vertex is paired with an unstarred one. Therefore, Z = 0 1s a necessary condition for the exi:stence of Kekule structures. However, th1s condition iis not sufficient and one can find benzenoid graphs with Z = 0 which do not possess Kekule structures. A trial and error procedure show:s that XI and XII are such graphs with minimum R value (R = 11). XII A sufficient condition for K > 0 is the existence -of a Hamiltonian path in the molecular graph. If N; = 0, the perimeter is a Hamiltonian cycle and therefore cata-condensed systems always possess Kekule structures. Unfortu-nately, there are benzenoid graphs with K > 0 having no Hamiltonian paths. XIII is the smallest example of this kind (R = 9). In concluding this section it i:s to be emphasized that there is no simple recipe to decide by inspection of the molecular graph whether K = 0 or not. In other words, the necessary and sufficient conditions for the existence of Kekule structures seem to be rather complicated. TOPOLOGICAL PROPERTIES XIIT A SPECTRAL PROPERTY OF BENZENOID GRAPHS The polynomial N P (x) = k crj xN - j j = O is called the characteristic polynomial of a graph2 if N P (X) = fl (X - Xj) j = l 213 (8) (9) The relations that exist between the structure of the molecular graph and the coefficients a.i of the characteristic polynomial (the Sachs theorem11) are extensively discussed elsewhere3,12. Thus, it is well known that for all graphs the following simple expressions are valid: N a1 = O or k xi = O j = l N a2 = - M or ~ .r.~ = 2 M j = l (10) (11) This also means that not only the values of N and M, but also the parameters R, N;, T and S can be calculated from the graph spectrum. Here we will be interested in a topological formula for a, . For benzenoid graphs which, of course, do not contain three- and four--membered cycles, a_, is equal to the number of choices of a pair of nonadja-cent edges11 ,12. This purely combinatorial problem can be solved using the following reasoning. Let us consider an edge (p, q) between the vertices p and q. There are M - DP - Dq + 1 edges which are not adjacent with (p, q). Therefore, 1 a4 = 2 k (M-Dp-Dq +l) (12) (p,q) where the summation goes over all edges. Of course, 214 I. GUTMAN ~ (M + 1) = M (M + l) (p,q) (13) Now, every vertex p appears DP times in the above summation and therefore N ~ (DP+ Dq) = ~ D~ (14) (p,q) p = l Thts finally gives 1 N a4 = - [M (M + 1) - ~ D~] 2 p = l (15) and 1since1a N N 4 a4 + a2 ~ x 2 + ~ x 4 = O J J (16) j = l j=l N N ~ xi = 2 ~ D~ -2 M (17) j = l p = l Using the equations (4) and (7), both relations (15) and (17) can be expressed in terms of the topological parameters Ni and R: 1 a4 = - [(5 R- N.)2 -11 R + N + 4] 2 I I N ~ xi = 6 (7 R - N; - 1) j = l (18) (19) The fact that the ex·pressicm (19) is equal to the number of closed walks of length four in the graph, and its consequences on the value of the total it-electron energy is discussed elsewhere14 • The simple relations which are presented in this note are only the first step in a complex topological study of benzenoid systems. Hope can be expres-sed that this work will stimulate further investigations in the same direction. Acknowledgement. The author would like to thank Dr. D. Cvetkovic (Beograd), Prof. N. Trinajstic (Zagreb), and Prof. C. F. Wilcox, Jr. (Ithaca) for useful suggestions and help in the preparation of the manuscript. REFERENCES 1. For review and further references see: I. Gutman ;md N. Trina j st i c, Fortschr. Chem. Forsch. 42/2 (1973) 49. 2. We use here the usual graph-theoretical nomenclature, which is defined in e. g. Refs. 1 and 3. 3. A. Gr a ova .c, I. Gutman, N. Trina j st i c, and T. Ziv k o vi c, Theor. Chim. Acta 26 (1972) 67; D. Cvetkovic, I. Gutman, and N. Trinajstic, Croat. Chem. Acta 44 (1972) 365. 4. F. Har a r y, Usp. Mat. Nauk 24 (1969) 179. 5. A. T. Ba 1 ab an and F. Har a r y , Tetrahedron 24 (1968) 2505; A. T. Ba 1 a-b a n, ibid. 25 (1969) 2949. 6. M. J . S. Dewar and H. C. Longuet-Higgins, Proc. Roy. Soc. Ser. A 214 (1952) 482. TOPOLOGICAL PROPERTIES 215 7. D. Cvetkovic, I. Gutman, and N. TrinajstiC, J. Chem. Phys. 61 (1974) in press. 8. G. G. Ha 11, Internat. J. Math. Educ. Sci. Technol. 4 (1973) 233; Inst. Math. Appl. Bull. 8 (1972) 226. 9. W. Eng 1 end and K. Rued en berg, J. Amer. Chem. Soc. 95 (1973) 8769. 10. D. H. Ro u v ray, Roy. Inst. Chem. Rev. 45 (1972) 6. 11. H. Sachs, Publ. Math. (Debrecen) 11 (1963) 119. 12. H. Hos o ya, Theor. Chim. Acta 25 (1972) 215. 13. Eq. (16) is a special case of the Newton identity. For a proof of this identity see: I. Gutman and N. Trina j st i c, Chem. Phys. Lett. 17 (1972) 537. 14. I. Gutman and N. Trina j st i c, Chem. Phys. Lett. 20 (1973) 257. SAZETAK 0 nekim topoloskim svojstvima benzenoidnih sustava Ivan Gutman De:5inirana je klasa grafova koja reprezentira benzenoidne sustave i izvedene su neke elementarne relacije koje vaZe u toj klasi. Diskutirana je egzistencija Keku-leovih struktura i ukazano na niz teskoea oko rjefavanja tog problema. Izvedena je jednadzba koja povezuje .spektar benzenoidnih grafova s nizom topoloskih para-metara tih grafova; ta jednadzba jedina je poznata relacija u toj klasi grafova. INSTITUT »RUDER BOSKOVI<'.:« 41000 ZAGREB Primljeno 27. svibnja 1974. 

Some Topological Properties of Benzenoid Systems

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