An algorithm for setting up adjacency matrices of benzenoid
hydrocarbons from their boundary sequences is detailed. A recipe
for computing the boundary code of a benzenoid structure from
its adjacency matrix is also given. Finally, an automatic procedure
is devised for computing approximate TRE values of benzenoids
from their boundary sequences via the Gutman-Petrovic formulae

Ganija Jashari

Jan V. Knop

Klaus Szymanski

Nenad Trinajstić

English

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CROATICA CHEMICA ACTA CCACAA 56 (3) 443-450 (1983) CCA-1390 YU ISSN 0011 1643 UDC 539.19 Original Scientific Paper The Connection Between the Adjacency Matrix and the Boundary Code of Benzenoid Hydrocarbons Jan V. Knop and Klaus Szymanski Computer Centre, The University of Diisseldorf, 4000 Diisseldorf, Federal Republic of Germany and Ganija Jashari and Nenad Trinajstic The Rugjer Boskovic Institute, P.O.B. 1016, 41001 Zagreb, Croatia, Yugoslavia Received August 30, 1982 An algorithm for setting up adjacency matrices of benzenoid hydrocarbons from their boundary sequences is detailed. A recipe for computing the boundary code of a benzenoid structure from its adjacency matrix is also given. Finally, an automatic procedure is devised for computing approximate TRE values of benzenoids from their boundary sequences via the Gutman-Petrovic formulae. INTRODUCTION In this work we wish to report a method for computing the adjacency matrix of benzenoid hydrocarbons from their boundary codes. The carbon skeletons of benzenoids will be depicted by graphs, called benzenoid graphs1•2 , which can be constructed in the plane by assembhng h regular hexagons in such a way that two hexagons have exactly one joint edge or are disjoint, and the covered area in the plane is simply connected3•4• This restriction excludes benzenoid hydrocarbons with a »hole« in the interior, e. g. corannu-lenes. Since benzenoid graphs represent networks of hexagons5•6, it is also customary to call them, for brevity, polyhexes7• Another name used for these structures is arenes8• In the present report we will use the term polyhex for a benzenoid hydrocarbon. Here we shall also show how the boundary code of a polyhex can be obtained from its adjacency matrix. The definitions of the boundary and the interior of a polyhex (benzenoid structure) are given in references 4 and 5. Finally, we shall describe a simple way to calculate approximate9 topological resonance energies2,10 ,11 of benzenoid hydrocarbons by using the Gutman-Pe-trovic formulae from eigenvalues of the adjacency matrix. * Presented at The IUP AC International Symposium on T !ieoretical Organic Chemistry, held in Dubrovnik, Croatia, August 30 - September 3, 1982. 444 J. V . KNOP ET AL. 1. Computing the Adjacency Matrix of a Polyhex from its Bondary Sequence In our earlier work12- 14 we made the statement tha:t the interior of a polyhex is constructable knowing the boundary because of the uniformity of the i:nterior. This statement, which becomes immediately clear on inspection of the polyhex, means that the boundary sequence12 of a polyhex includes complete information about the structure of the polyhex. On the other hand the adjacency matrix of a polyhex also contains complete information about its structure. Therefore it must be possible to compute the adjacency matrix from the boundary sequence. The adjacency matrix is needed for studying various properties of polyhexes, such as spectra. Consider the following polyhex denoted by G. G To construct the adjacency matrix of G we depict this polyhex in the (x, y)--plane in a very uncommon way: Since the length of an edge in the plane is defined to be unity, in the above representation of G the vertices, V (G), can be described as a subset of the Cartesian product of the integers Z, V(G) c Z X Z Our first goal is to find the vertex-set V (G) in Z X Z. Let the polyhex be given in the boundary notation12 as a sequence of edges, The mapping f from the set of edge types onto the cartesian product of integers, is defined by, BENZENOID HYDROCARBONS f : { 1, 2, • .. , 6}-+ z x z f: 1-+ (0,1) 2-+ (1,0) 3-+ (1,0) 4-+ (0,-1) 5-+ (-1,0) 6-+ (-1,0) 445. The set of boundary vertices of the polyhex in Z X Z can be computed as,. VB : ={vi: vi:= ~ f(ei), l~i~k}cV(G)cZXZ j=1 In this construction of the polyhex two vertices V : ~ (v1, v2) € V {G) W : = (w1, w2) € V (G) are adjacent if (i) v, w € VB and there is an integer i that i-1 i w = ~ f (ei) and v = ~ f (ei) = w + f (e) j=l }=1 (ii) V2 = W2 and J v 1 -w1 J = 1 and valency v +valency w > 4; V1 = W1 and I V2-w2 J = 1 and J v1-u1 I mod 2 = J V2-u2 J mod 2 and valency v + + valency w > 4 with u2 : =max { u/: (u/, u/) €VB}, u1 : = min{u/: (u{,Ui') €VB}. Note the following properties of the vertices of polyhexes:4,5 (1) If a vertex has valency 2, it is a boundary vertex. (2) Some vertices on the boundary have valency 3. They represent points at which the interior of polyhex is connected. (3) All vertices of the interior have valency 3. (4) There are only two types of valency 3 vertices possible in our construction, _J_- • T but in both cases it is true that if, v = (v 1, v2) € V (G) and valency v = 3 then (V1 + 1, V2), (V1 - 1, V 2) € V (G), This important fact gives us the possibility of constructing all vertices of V (G). Let v = (vi, v2) € V (G)""'-VB be an inner vertex of the polyhex, then there exists a finite sequence of adjacent valency 3 vertices of V (G), 446 J. V. KNOP ET AL. v0 = v, v(1) : = (v1 + 1, v2) , •• ., v<n> : = (v1 + n, v 2) with v <0 > E VB 'That means each vertex of the interior can be reached from a valency 3 vertex of the boundary by a path of the special type as above. At this point we can make use of the successor mapping s and the proces-:sor mapping p12, 1~2 1~6 2~3 2~1 s: 3~4 p: 3~2 4~5 4~3 5~6 5~4 6~1 6~5 Qn the boundary of a polyhex an edge ei po~nts to a vertex of valency 3 if ei+I = p (ei) (1 :::;:; i :::;:; k - 1), e1 = p (ek) .and points to a vertex of valency 2 if 'The set ' VB3 : ={vieVB:vi= ~ f(ei) and ei+i=p(ei), 1:::;:;i::=;:;k} j=1 is the set of all valency 3 vertices on the boundary. Starting with a vertex v = (vi. v 2) E VB3 we compute all v<n> = (v; + 1, v 2) until v<n> E VB. Thus we ge-nerate all boundary vertices and all inner vertices. As we know from (i) and (ii) which vertices are adjacent it is easy to construct the adjacency matrix. We find canonical labels by ordering the set -0f vertices in the following way, v = (V1, Vz) , w = (W1, Wz) € v (G) v < w if v 2 > w 2 or v 2 = w 2 and v 1 < w 1• A FORTRAN program was written to generate the adjacency matrix using the bo'll!ndary notation of the polyhex. This program may be obtained on request from the authors. 2. Computing the Boundary Code of a Polyhex from its Adjacency Matrix Here we develop an algorithm to compute the boundary code of a poly-hex from its adjacency matrix. Let a polyhex be labeled in an arbitrary way and let A be the adjacency matrix set up according to this labeling. Let v E V (G) be a vertex and V (G) the vertex-set of the polyhex G. By h (v) we denote the number of different hexag-0ns containing v . Of course, 1 :::;:; h (v) :::;:; 3 for all v e V (G) The vertex set of the boundary of a polyhex can be characterized as VB= {v € V(G) : h (v) ~ 3}. BENZENOID HYDROCARBONS 447 Let t be the number of the terminal vertices equal to paths of lenght 6 start-ing at v, then, h (v) = Y. t (v) for all v E V (G). So VB can be determined as VB= {v E V(G) : y. t(v) ¢ 3}. We notice here that all vertices of valency v = 2 are elements of VB, i.e. -elements of the boundary. Let v0 be an element of VB, v 0 E VB, with valency v0 = 2. Because of the valency 2 both neighbours of v 0 are elements of the boundary. Let v1 be one -0f them. We consider two cases: {l) Let the valency of v1 be 2. From A we find the neighbour V u >"' v 0. Because of the valency 2 of vi, v 11 must be a boundary vertex. {2) Let the valency of v 1 be 3. From A we find the neighbours Vu >"' Vo and v 12 >"' v0• If one of them has valency 2, it is the next vertex on the boundary. If both of them have valency 3, we compute all paths of length 5 starting at vu and count the number of terminal vertices equal to V1. The number of terminal vertices equal to v 1 is the number of different hexagons con-taining v 1 and vu. Since v 1 E VB, vu is a boundary vertex if this number is 1. If this number is not 1, v 12 is a boundary vertex. By this algorithm we determine the second neighbour of v 1• Now con-:sidering v 1 as v 0, we start the algorithm once again and compute with L (the length of the boundary) steps the boundary sequence v 0, vi, .. . , L - 1 for the .given polyhex. The code of the boundary sequence is obtained by, code (v0) = 1 { s (code (v;)) if valency v ; = 2 . code (v;+1l = p (code (v;)) if valency vi = 3 O ~ i ~ l - 1 where s is the ·successor mapping and p the processor mapping. :3 . A Remark on Computing the Approximate Topological Resonance Energy of Benzenoid Hydrocarbons Gutman and Petrovic9 have recently proposed two formulae for the .approximate computation of topological resonance energies, TRE's, of ben--zenoid hydrocarbons. The TRE theory is well described elsewhere2,9- 11,15, so we do not need to detail it here. The Gutman-Petrovic formulae depends on only two topological para-meters : the number of hexagons h and the number of Kekule structures K of .a benzenoid hydrocarbon. These formulae a:re given below, TRE (1) = 1.7101 h 116 (K116 -1) (K - lf116 + 0.040 a nd TRE (2) = 0.2614 h 116 (ln K)"6 + 0.077 448 J . V. KNOP ET AL. The essential quantity for both formulae is the number of Kekule structures. There are numerous ways to evaluate K of polyhexes15•16 One way is from the value of the am coefficient of the characteristic polynomial, P (G; x), of G, K = ( 11 am I )'/• where m is the number of vertices in G. The value of am may be easily obtained since we can set up the adjacency matrix of a polyhex by 1:he algo-rithm described in section 2 and then can compute the eigenvalues of this matrix Xi, x2, • • • , Xm by a suitable programme (F cf) AAF of NAG). Thus we obtain, i , I ! m A FORTRAN program is written tha1: produces approximate TRE values of benzenoids from their boundary codes. For most benzenoid hydrocarbons is much easier to set up the boundary sequence than the adjacency matrix. This is the reason why we 'start our procedure from the boundary sequence and not from the adjacency matrix. Therefore, our procedure consists of the fol-lowing steps: 1. boundary sequence ~ 2. adjacency matrix ~ 3. set of eigenvalues ~ 4. number of Kekule structures ~ 5. TRE values A numerical example is given in the Table TABLE Example of Obtaining the Approximate TRE Values from the Boundary Code fo r a Given Benzenoid Hydrocarbon G G 1. Boundary Sequence (65654323456561612123216121234543234545) Star denotes the bond at which the boundary sequence starts. BENZENOID HYDROCARBONS :2. Adjacency Matrix (copy of the computer output) 0100100000000000000000000000000000000000 1010000000000000000000000000000000000000 0100001000000000000000000000000000000000 0000100100000000000000000000000000000000 1001010000000000000000000000000000000000 0000101001000000000000000000000000000000 0010010000000000000000000000000000000000 0001000010000000000000000000000000000000 0000000101000000100000000000000000000000 0000010010100000000000000000000000000000 0000000001010000001000000000000000000000 0000000000101000000000000000000000000000 0000000000010000000010000000000000000000 0000000000000010000000100000000000000000 0000000000000101000000000000000000000000 0000000000000010100000001000000000000000 0000000010000001010000000000000000000000 0000000000000000101000000010000000000000 0000000000100000010100000000000000000000 0000000000000000001010000000100000000000 0000000000001000000100000000000000000000 0000000000000000000000100000010000000000 0000000000000100000001010000000000000000 0000000000000000000000101000000100000000 0000000000000001000000010100000000000000 0000000000000000000000001010000000000000 0000000000000000010000000101000000000000 0000000000000000000000000010100000000000 0000000000000000000100000001000000000000 0000000000000000000001000000001000000000 0000000000000000000000000000010100010000 0000000000000000000000010000001010000000 0000000000000000000000000000000101000100 0000000000000000000000000000000010100000 0000000000000000000000000000000001000001 0000000000000000000000000000001000001000 0000000000000000000000000000000000010100 0000000000000000000000000000000010001010 0000000000000000000000000000000000000101 0000000000000000000000000000000000100010 3. Set of Eigenvalues -2.6660 -2.5096 -2.2942 -2.1995 -1.9706 -1.7207 -1.5023 -1.4673 -1.1633 -1.0532 -1.0000 -0.5827 -0.3700 0.3700 0.8442 1.0000 1.0532 1.4355 1.4673 1.5023 2.1995 2.2942 2.5096 4. Number of Kekule Structures K = 109 5. TRE Values TRE (1) = 1.404 TRE (2) = 1.468 -1.4355 -1.3075 -0.8442 -0.7110 0.5827 0.6446 1.1633 1.2195 1.7207 1.8464 2.6660 449 -1.8464 -1.2195 -0.6446 0.7110 1.3075 1.9076 450 J. V. KNOP ET AL. CONCLUSIONS In this work we have presented an algorithm that can be used for setting up the adjacency matrices .of (large) benzenoid hydrocarbons from the bound-ary sequences12, which numerically represent benzenoids and are easily avail-able. The algorithm can be straightforwardly programmed in FORTRAN. However, if we have already obtained in some way the adjacency matrix of the benzenoid hydrocarbon, it is shown how the boundary code of a polyhex can be obtained from it. The boundary codes may be used in several ways14• For example, the boundary codes may be employed for counting isomeric benzenoid structures. A procedure based on this idea is already developed1z-14• We have also shown how the Gutman-Petrovic formulae for approximating the TRE of benzenoids may be automatically computed from their adjacency matrices, which in turn may be set up from the corresponding boundary sequences, which, as it was shown, fully determine the benzenoid structures. Acknowledgements. - This work was supported in part by the German-Yugo-slav scientific cooperation administrated through Internationales Bilro-KFA-Jtilich. We also thank the referees for their valuable and extensive comments. REFERENCES 1. I. Gutman, Croat. Chem. Acta 46 (1974) 209. 2. A. Gr a ova c, I. Gutman, and N. Trina j st i c, Topological Approach to the Chemistry of Conjugated Molecules, Lecture Notes in Chemistry Vol. 4, Springer, Berlin 1977. 3. F. Har a r y, Graph Theory, Addison-Wesley, Reading, Mass., 1972 (third printing). 4. D. C vet k o vi c, I. Gutman, and N. Trina j st i c, J. Chem. Phys. 61 (1974) 2700. 5. B. D z o n o v a - J e r m a n - B 1 a z i c and N. T r i n a j s t i c, Computers and Chemistry 6 (1982) 121. 6. B. D z o n o v a - J e rm a n - B 1 a z i c and N. T r i n a j s t i c, Croat. Chem. Acta 55 (1982) 347. 7. A. T. Ba 1 ab an, in: Chemical Applications of Graph Theory, Edited by A. T. Ba 1 ab an, New York 1967, p. 63. 8. A. T. B a 1 ab an and F. Har a r y, Tetrahedron 24 (1968) 2505. 9. I. Gutman and S. P et r o vi c, Bull. Soc. Chim. (Beograd) 46 (1981) 459. 10. I. Gutman, M. Mi 1 u n , and N. Trina j st i c, Math. Chem. (Milhlheim/ !Ruhr) 1 (1975) 175. 11. I. Gutman, M. Milun, and N. Trinajstic, J. Amer. Chem. Soc. 39 (1977) 1692. 12. J . V. Knop, K. S z y mans k i, Z. Jeri c e vi c, and N. Trina j st i c, J. Comput. Chem., in press. 13. J. V. Knop, K. S z y mans k i, Z. J er ice vi c, and N. Trina j st i c, Int. J. Quantum Chem., in press. 14. N. Trina j st i c, Z. Jeri c e v iC, J . V. Knop, W. R. Mil 11 er, and K. S z y mans k i, Pure Appl. Chem., in press. 15. N. Trina j st i c, Chemical Graph Theory, CRC, Boca Raton, in press. 16. G. J a sh a r i, M. Sc. T hesis, University of Zagreb, 1982. SAZETAK Odnos izmedu matrice susjedstva i rubnog koda benzenoidnih ugljikovo'dika J. V. Knop, K. Szymanski, G. Jashari, i N. Trinajstic Prikazan je algoritam za nalazenje matrice susjedstva benzenoidnih ugljiko-vodika iz njihovih rubnih nizova. Opisan je, takoder, algoritam za raeunanje rubnog koda benzenoidne strukture iz njezine matrice susjedstva. Na kraju, izvedena je automatska procedura za racunanje vrijednosti TRE benzenoida iz njihovih rubnih nizova s pomocu formule Gutmana i Petrovica. 

The Connection Between the Adjacency Matrix and the Boundary Code of Benzenoid Hydrocarbons

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