Iranian Journal of Mathematical Chemistry, Vol. 5, No.2, November 2014, pp

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1 Iranan Journal of Matematcal Cemstry, Vol. 5, No.2, November 204, pp IJMC Altan dervatves of a grap I. GUTMAN (COMMUNICATED BY ALI REZA ASHRAFI) Faculty of Scence, Unversty of Kragujevac, P. O. Bo 60, Kragujevac, Serba, and State Unversty of Nov Pazar, Nov Pazar, Serba ABSTRACT. Altan dervatves of polycyclc conjugated ydrocarbons were recently ntroduced and studed n teoretcal organc cemstry. We now provde a generalzaton of te altan concept, applcable to any grap. Several earler notced topologcal propertes of altan dervatves of polycyclc conjugated ydrocarbons are sown to be te propertes of all altan dervatves of all graps. Among tese are results pertanng to Kekulé structures/perfect matcngs, determnant of te adjacency matr, and grap spectrum. Keywords: altan grap, molecular grap, Kekulé structure, perfect matcng, spectrum (of grap). INTRODUCTION Altan dervatves of polycyclc conjugated molecules recently came nto te focus of attenton of teoretcal organc cemsts [-5]. Te name altan s an abbrevated form of alternatng annulene, wc s a fragment encrclng te parent conjugated system. Te altan dervatve of a conjugated ydrocarbon s constructed so tat eac ydrogen atom s replaced by a vnyl group, and eac two adjacent vnyl groups are condensed nto a new cycle. Te constructon of altan-penantrene s sown n Fg.. An nterestng and attractve property of altan molecules s tat te π-electron system of te annulene tat surrounds te parent ydrocarbon s only weakly nteractng wt te π- electron system of te parent ydrocarbon. From te pont of vew of Kekulé structures, te π- electron conjugaton n te annulene and te parent ydrocarbon are fully ndependent. In partcular, te Kekulé structure count of te altan dervatve s equal to te product of te Kekulé structure count of te parent ydrocarbon and te number of Kekulé structure of te annulene (wc, of course s equal to two) [5,6]. Emal: gutman@kg.ac.rs Receved August 24, 204; Accepted August 25, 204.

2 86 IVAN GUTMAN Fg.. Constructon of altan-penantrene from penantrene. Note tat te permeter of altan-penantrene s a [20]annulene, encrclng te penantrene subunt. It s easy to see tat f te number of ydrogen atoms of te parent conjugated molecule s even, ten te annulene n te altan-dervatve as sze 4k+2 (and tus contans 4k+2 π-electrons). Accordng to te Hückel (4k+2)-rule [7-9], suc a π-electron system contrbutes to antaromatcty and possesses a paratropc rng currect. In wat follows, we sow tat te above lsted propertes of altan-molecules are propertes of a muc more general class of grap dervatves, wc we defne n te subsequent secton. 2. GENERALIZING THE CONCEPT OF ALTAN -DERIVATIVES Defnton. Let G be a grap of order n. For 2 n, let U ( v, v2., v ) be an ordered -tuple of vertces of G. Te altan dervatve of te grap G wt regard to U s te grap G G ( U ) constructed n te followng manner. For, 2,,, attac a new verte to te verte v. Ten attac a new verte y to te verte. Ten for, 2,,, connect te vertces y and and also connect te vertces y n and. As a drect consequence of Defnton we see tat: ) If te grap G as n vertces, ten G G ( U ) as n+2 vertces. 2) If te grap G as m edges, ten G G ( U ) as m+3 edges.

3 Altan dervatves of a grap 87 3) Te degree of te vertces, 2,, s 3. 4) Te degree of te vertces y, y2,, y s 2. Remark 2. If G s a molecular grap of a polycyclc conjugated ydrocarbon, ten n ts altan-dervatve G G ( U ), te vertces U ( v, v2., v ) are tose correspondng to carbon atoms to wc a ydrogen atom s connected, arranged along te permeter. Ten s just te number of ydrogen atoms of te underlyng ydrocarbon. If so, ten te cycle formed by te vertces y2 y2 y corresponds to te annulene encrclng te parent ydrocarbon. Evdently, ts sze s 2. In wat follows, we denote by by te vertces y2 y2 y. 2 te (2) membered cycle of G G ( U ), nduced 3. ON PERFECT MATCHINGS OF ALTAN GRAPHS If u and v are two adjacent vertces of a grap, ten te edge connectng tem wll be denoted by uv. We say tat uv covers te vertces u and v. Two edges of a grap are sad to be ndependent f tey cover 4 vertces. A perfect matcng of a grap G s a set of mutually ndependent edges wc cover all vertces of G. Te number of perfect matcngs of te grap G wll be denoted by K(G). In case of molecular graps, perfect matcngs are n a one-to-one correspondence wt Kekulé structures. Ten K(G) s just te Kekulé structure count of te underlyng conjugated molecule. For detals on ts correspondence see [8,0]. Teorem 3. Let te vertces of te altan grap edges v,, 2,,, do not belong to any perfect matcng of Proof. Suppose te contrary, namely tat M ( G ) G be labeled as n Defnton. Ten te G. M ( G ) s a perfect matcng of v. Ten, n order tat te verte y be covered, t must be y n order tat te verte y 2 be covered, t must be y 2 3 M ( G ) we see tat n order tat te verte y be covered, t must be owever, te verte y cannot be covered n, and te edges v and y are already n 2 M ( G ) G and tat. Ten,. Contnung ts argument, y M ( G ). Ten, M ( G ) because y s adjacent only to and M G ( ). Terefore possble, and analogously v M ( G ) s not possble for any, 2,,. v M ( G ) s not Corollary 4. For any grap G and for any -tuple U, K G U ( ( )) 2 K ( G). Proof. By Teorem 3, deletng te edges v from G does not cange te number of perfect matcngs. After deletng all v,, 2,,, wat remans s te dsconnected subgrap G 2. Corollary 4 follows from K G 2 K G K 2 ( ) ( ) ( ) and K( 2 ) 2.

4 88 IVAN GUTMAN Remark 5. Te Kekulé-structure equvalents of Teorem 3 and Corollary 4, vald for altandervatves of conjugated ydrocarbons, were notced n all papers [-4]. A formal proof tereof, applcable to altan-benzenods was offered n [5]. We now see tat tese are just specal cases of a muc more general regularty. 4. DETERMINANT OF ADJACENCY MATRIX OF ALTAN GRAPHS Let G be a grap of order n. A Sacs grap of G s a subgrap of G wose all components are cycles and/or 2-verte complete graps K 2 [8,,2]. Denote by S( G ) te set of all Sacs graps tat are spannng subgraps of te grap G (.e., tat ave same number of vertces as G). It s known [8,-3] tat te determnant of te adjacency matr A( G ) of a grap G satsfes te relaton n p( ) c( ) det A( G) ( ) ( ) 2 () S ( G) were p( ) and c( ) are, respectvely, te number of components and of cycles n te Sacs grap. Teorem 6. Let te vertces of te altan grap edges v,, 2,,, do not belong to any Sacs grap from te set G be labeled as n Defnton. Ten te S( G ). Proof. Suppose te contrary, namely tat tere ests a Sacs grap S( G ), suc tat te edge v belongs to one of ts components. We ave to dstngus between two cases: Eter (a) v belongs to a K2-component of, or (b) v belongs to a cycle of. Case (a). If v belongs to a K2-component of, ten also y2 must belong to anoter K2-component of. Same olds for y2 3,, y Ten, owever, te verte y cannot belong to any component of, because y s adjacent only to and, and te edges v and y are already n. Terefore te edge v cannot belong to a K2 - component of, 2,,., 2 S( G ), and analogously, te same olds for te edges v for all Case (b). It te edge v belongs to a cyclc component of, ten eter te vertces y or te vertces y, (but not bot pars!) belong to te same cycle. Wtout loss of generalty, assume ts s te par y,. Ten te edge y 2 must belong to a K2 - component of. Same olds for te edges y2 3,, y 2. Ten, owever, te verte y cannot belong to any component of, because y s adjacent only to and, and tese latter vertces are already n. Terefore te edge v cannot belong to a cyclc component of, 2,,. S( G ), and analogously, te same olds for te edges v for all

5 Altan dervatves of a grap 89 Corollary 7. For any grap G and for any -tuple U, det A( G ( U )) det A( G) det A( 2). Proof. By Teorem 6, and bearng n mnd Eq. (), deletng te edges v from G does not cange te determnant of te adjacency matr. After deletng all v,, 2,,, wat remans s te dsconnected subgrap det A( G ) det( G) det( ). 2 2 G 2. Corollary 7 follows now from As well known [8,,2], and as a drect consequence of Eq. (), te determnant of te cycle of sze 2 s equal to -4 s s odd, and s equal to 0 s s even. Ts yelds: Corollary 8. For any grap G and for any -tuple U, f s even, ten wereas f s odd, ten det A( G ( U )) 4det A( G). det A( G ( U )) 0 Corollary 9. For any grap G and for any -tuple U wt even, te altan-grap sngular,.e., ts spectrum possesses at least one zero egenvalue. G s Remark 0. Polycyclc conjugated ydrocarbons necessarly possess an even number of ydrogen atoms. Terefore, n all cemcally relevant altan-dervatves, te parameter s even, mplyng tat one of te grap egenvalues s equal to zero. In te language of teoretcal cemstry [8,,4], all altan-dervatves of conjugated ydrocarbons ave a nonbondng molecular orbtal. Ts mportant fact was observed n all earler papers [-5]. We now see tat ts s just specal cases of a muc more general regularty. Te nullty ( G) of a grap G s te multplcty of ts egenvalue zero [4]. Corollary. If s even, ten ( G ). Te nullty of G may be greater tan unty. In partcular, ( G ) ( G), wereas te nullty of te parent grap G may assume any value between 0 and n [4]. Te case ( G ) ( G) s, for eample, encountered f G s te grap wtout edges. Te case ( G ) ( G) s encountered f G s non-sngular. If s odd, ten te stuaton wt nullty s smple: ( G ) ( G) olds n all cases.

6 90 IVAN GUTMAN REFERENCES. G. Monaco and R. Zanas, On te addtvty of current densty n polycyclc aromatc ydrocarbons, J. Cem. Pys., 3 (2009), G. Monaco, M. Memol and R. Zanas, Addtvty of current densty patterns n altanmolecules, J. Pys. Org. Cem., 26 (203), T. K. Dckens and R. B. Mallon, Topologcal Hückel-London-Pople-McWeeny rng currents and bond currents n altan-corannulene and altan-coronene, J. Pys. Cem., A 8 (204), T. K. Dckens and R. B. Mallon, π-electron rng-currents and bond-currents n some conjugated altan-structures, J. Pys. Cem., A 8 (204) I. Gutman, Topologcal propertes of altan-benzenod ydrocarbons, J. Serb. Cem. Soc. 80 (205), G. Monaco and R. Zanas, On te addtvty of current densty n polycyclc aromatc ydrocarbons, J. Cem. Pys.,3 (2009), G. M. Badger, Aromatc Caracter and Aromatcty, Cambrdge Unv. Press, Cambrdge, A. Graovac, I. Gutman and N. Trnajstć, Topologcal Approac to te Teory of Conjugated Molecules, Sprnger, Berln, R. C. Haddon, Unfed teory of resonance energes, rng currents and aromatc caracter n (4n+2)-π electron annulenes, J. Am. Cem. Soc.. 0 (979), D. Cvetkovć, I. Gutman and N. Trnajstć, Kekulé structures and topology, Cem. Pys. Lett., 6 (972) I. Gutman and O. E. Polansky, Matematcal Concepts n Organc Cemstry, Sprnger, Berln, A. Graovac, I. Gutman, N. Trnajstć and T. Žvkovć, Grap teory and molecular orbtals. Applcaton of Sacs Teorem, Teor. Cm. Acta, 26 (972), A. Graovac and I. Gutman, Te determnant of te adjacency matr of a molecular grap, MATCH Commun. Mat. Comput. Cem. 6 (979), I. Gutman and B. Borovćann, Nullty of graps: An updated survey, n: D. Cvetkovć and I. Gutman (Eds.), Selected Topcs on Applcatons of Grap Spectra, Mat. Inst., Belgrade, 20, pp