Bond Additive Modeling 5. Mathematical Properties of the Variable Sum Exdeg Index

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1 CROATICA CHEMICA ACTA CCACAA ISSN 00-6 e-issn -7X Crot. Chem. Act 8 () (0) 9 0. CCA-5 Orgl Scetfc Artcle Bod Addtve Modelg 5. Mthemtcl Propertes of the Vrble Sum Edeg Ide Dmr Vukčevć Fculty of Nturl Sceces d Mthemtcs Uversty of Splt Nkole Tesle HR-000 Splt Crot (E-ml: vukcev@pmfst.hr) RECEIVED JANUARY 00; REVISED JULY 00; ACCEPTED DECEMBER 00 Abstrct. Recetly dscrete d vrble Adrtc dces hve bee troduced d t hs bee show tht the sum α -edeg de s good predctor (whe vrble prmeter s equl to 0.7 ) of the octol-wter prtto coeffcet for octe somers. Here we study mthemtcl propertes of ths descrptor. Nmely we lyze etreml grphs of ths descrptor the followg clsses: the clss of ll coected grphs the clss of ll trees the clss of ll ucyclc grphs the clss of ll chemcl grphs the clss of ll chemcl trees the clss of ll chemcl ucyclc grphs the clss of ll grphs wth gve mml degree the clss of ll grphs wth gve mml degree the clss of ll trees wth gve umber of pedt vertces d the clss of ll coected grphs wth gve umber of pedt vertces. Also my ope problems bout vrble Adrtc dces re proposed. (do: 0.556/cc667) Keywords: moleculr descrptor etreml vlues etreml grphs chemcl grphs ucyclc grphs trees de c be rewrtte s: INTRODUCTION Recetly dscrete d vrble Adrtc dces hve bee troduced d studed. Predctve d mthemtcl propertes of dscrete Adrtc dces hve bee lyzed ppers. Predctve propertes of vrble Adrtc dces hve bee studed Ref.. It hs bee foud tht three vrble Adrtc dces hve especlly good predctve propertes mely: ) the verse sum -.95-deg de s well correlted wth stdrd ethlpy of formto of octe somers R 0.75 ) the verse sum 0.-lodeg de s well correlted wth totl surfce re of octe somers R 0.9 ) the sum 0.7-edeg de s well correlted wth the octol-wter prtto coeffcet R I ths pper we restrct our tteto to the vrble sum edeg de. The vrble sum edeg de s defed by: SEI G uv E G du dv 0. where E G s the set of edges of G. Note tht ths SEI G du dv uv E G du u V G v V G :uv E G u V G d u du where V G s the set of vertces of G. Hece ths de c be cosdered s sum of verte cotrbutos such tht the cotrbuto of ech verte depeds solely o ts degree. Oe mmedtely see prllel to the well kow the frst Zgreb de defed by: M G u V G du. The mthemtcl d predctve propertes of Zgreb de hve bee etesvely studed (see Refs. 6 d refereces wth d for recet mthemtcl studes Refs. 7 ). I ths pper we lyze grphs wth etreml vlues of the SEI de the followg clsses: the clss of ll coected grphs the clss of ll trees the clss of ll ucyclc grphs the clss of ll chemcl grphs the clss of ll chemcl trees the clss of ll

2 9 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide chemcl ucyclc grphs the clss of ll grphs wth gve mml degree the clss of ll grphs wth gve mml degree the clss of ll trees wth gve umber of pedt vertces d the clss of ll coected grphs wth gve umber of pedt vertces. These results c be used for the detecto of chemcl compouds tht my hve desrble propertes. Nmely f oe fds some property well-correlted wth ths descrptor for some vlue of α the etreml grphs should correspod to molecules wth mml or mml vlue of tht property. Sce oe such property hs lredy bee foud ths my ecourge the further study of ths de. MATHEMATICAL PROPERTIES OF SUM - EXDEG INDEX FOR I ths pper we cosder oly smple coected grphs so from ow o by grph we mply smple coected grph. The umber of vertces of G wll be deoted by G d umber of edges by mg. By G we deote the umber of vertces of degree. Let f : be the fucto defed by f where 0. It c be esly see tht: Lemm. Fucto f s cresg fucto for ech. Proof: Iequlty f ' l 0 mples the Lemm. Lemm. Fucto f s cove fucto for ech. Proof:Iequlty f '' l l 0 mples the Lemm. From the Lemm t drectly follows tht: Proposto. SEI G for every grph G wth vertces d for ech. Equlty holds f d oly f G s complete grph. Proposto. SEI G for every grph G wth vertces d mml degree ; d for ech. Equlty holds f d oly f G s -regulr grph. Proposto 5. SEI G for every chemcl grph G wth vertces d for ech. Equlty holds f d oly f G s -regulr grph. Proposto 6. δ δ SEI δ G δ δ δ for every grph G wth vertces d mml degree δ ; d for ech. Equlty holds for the lower boud f d oly f G s δ -regulr grph. Equlty holds for the upper boud f d oly f G hs verte u of degree δ d G u s complete grph o vertces. Let us prove: Lemm 7. Let be umbers such tht... d let. The f. The equlty holds for the lower boud f d oly f the followg multsets re equl... d the equlty holds for the upper boud f d oly f.... Proof: Frst let us prove the lower boud. It c be esly see tht equlty holds for.... Let... be the -tuple wth the smllest vlue Suppose to the cotrry tht of f..... Wthout loss of geerlty we my ssume tht.... It c be esly see tht d tht but the... f f f f f f f f f f f becuse f s cove. Ths s cotrdcto wth the choce of.... Now let us prove the upper boud. It c be esly see tht the equlty holds for.... Let hghest vlue of f.... be the -tuple wth the Suppose to the cotrry tht.... Wthout loss of geerlty we my ssume tht.... It c be esly see tht d tht but the... f f f f f f f f f f f becuse f s cove. Ths s cotrdcto wth the choce of.... From here t drectly follows tht: Crot. Chem. Act 8 (0) 9.

3 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide 95 Proposto 8. SEI G for ech tree G wth vertces d for ech. The equlty for the lower boud holds f d oly f G s pth. The equlty holds for the upper boud holds f d oly f G s str S. Proof: Note tht du for ech verte u V G d tht du. uvg Let us recll tht every grph G cots spg tree T.e. subgrph T whch s tree such tht V G V T. Let us prove: Lemm 9. Let G be grph d T ts spg tree the SEIG SEIT for ech. Proof: Sce the degree of every verte T s ot lrger the G the Lemm follows. From Proposto 8 d Lemm 9 t follows: Corollry 9. SEI G for ech grph G wth vertces d for ech. The e- qulty for the lower boud holds f d oly f G s pth. Corollry 0. SEI G for ech chemcl grph G wth vertces d for ech. The equlty for the lower boud holds f d oly f G s pth. Deote by A set of ll multsets... such tht: ) ; )... ; ) t lest three umbers re greter the. Let us prove: Lemm. Let.... A The f. The equlty holds for the lower boud f d oly f... d the equlty for the upper boud holds f d oly f.... Proof: Frst let us prove the lower boud. It c be esly see tht equlty holds for.... Let... A be the -tuple wth smllest vlue of f. Suppose to the cotrry tht.... Wthout loss of geerlty we my ssume tht.... It c be esly see tht d tht but the:... f f f f f f f f f f f becuse fucto f s cove. From the choce of... t follows tht... A.e. tht t hs t lest elemets lrger the but ths s ot possble. Now let us prove the upper boud. It c be esly see tht equlty holds for.... Let A wth hghest vlue of f. tht... be the -tuple Suppose to the cotrry.... Wthout loss of geerlty we my ssume tht.... It c be esly see tht d tht but the... f f f f f f f f f f f becuse f s cove. From the choce of... t follows tht... A.e. tht t hs less the elemets lrger the. Ths s possble oly f... d but the smple clculto gves.e.... whch s cotrdcto. Let S be the grph obted from the str S by ddg edge coectg two leves s preseted the Fgure. Fgure. Grph A. Crot. Chem. Act 8 (0) 9.

4 96 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide From Lemm t drectly follows tht: Proposto. SEI G for every ucyclc grph G wth vertces d for ech. E qulty for the lower boud holds f d oly f G s cycle C. Equlty for the upper boud holds f d oly f G S. Proof: Deote V v... v d ote tht: ) dv... d... ; v ) dv... d ; v ) t lest three umbers d v re greter the. Let S k be clss of grphs wth vertces obted from Sk by replcg some edges by pths. Smlrly s bove t c be proved tht: Lemm. Let d let be umbers such tht... d let. The f..... From ths Lemm t follows: Equlty holds f d oly f Proposto. SEI G for every grph G wth vertces d mml degree d for ech. Equlty holds f d oly f G S k Proof: From Lemm 9 t follows tht t s suffcet to prove the clm for trees. Hece let us ssume tht G s tree. Deote V G v... v where dv. It s suffcet to ote tht dv... d v stsfy the codto of the prevous Lemm. Moreover the equlty holds f d oly f dv... d v but ths s possble f d oly f G S k. Usg smlr techques s bove t c be proved tht: Proposto 5. It holds tht k k k k SEI G k k k for every tree T wth vertces d k pedt vertces d for ech. The equlty for the lower boud holds f d oly f G s grph wth k vertces of degree k vertces of degree d k vertces of degree. The equlty for the upper boud holds f d oly f G S k. Proposto 6. It holds for every grph G wth vertces d k pedt vertces d for ech. The equlty for the lower boud holds f d oly f G s grph wth k vertces of degree k vertces of degree d k vertces of degree. The equlty holds for the upper boud f d oly f ll pedt vertces re djcet to the sme verte d the subgrph obted by elmto of pedt vertces s complete grph Proposto 7. SEI G for ech chemcl tree G wth vertces d for ech. The equlty holds for the lower boud holds f d oly f G s pth. The equlty holds for the upper boud f d oly f G hs oly vertces of degree d. Proposto 8. SEI G for ech chemcl ucyclc grph G wth vertces d for ech. The equlty holds for the lower boud holds f d oly f G s cycle. The equlty holds for the upper boud f d oly f G hs oly vertces of degree d.. MATHEMATICAL PROPERTIES OF SUM - EXDEG INDEX FOR Here we restrct our tteto to chemcl grphs. Frst let us lyze chemcl trees. Theorem 9. Let 0 d G be chemcl tree wth vertces. It holds: f 0 ; f 5 π 7 SEI G cos ; 8 f 5 π 7 cos ; 8 k. k k k SEI G k k Crot. Chem. Act 8 (0) 9.

5 97 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide f 0 cos ; SEI G. f cos Moreover the equlty the lower bouds holds f d oly f oe of the followg holds: 0 d G s pth P ; b) d G oly hs vertces of degrees d ; 5 π 7 d G oc) cos 8 ly hs vertces of degrees d ; 5 π 7 d G oly hs d) cos 8 vertces of degrees d ; 5 π 7 d G oly e) cos 8 hs vertces of degrees d ; ) the equlty the upper bouds holds f d oly f oe of the followg holds: 0 cos d G oly hs vertces of degrees d ; d G oly hs vertces of deb) cos grees d ; c) cos d G s pth P. ) Proof: Note tht G G G. Hece Moreover the equlty for the upper boud holds f d oly f: ) G 0 or m ; ) G 0 or m ; ) G 0 or m ; d the equlty for the lower boud holds f d oly f: I) G 0 or m ; II) G 0 or Crot. Chem. Act 8 (0) 9.

6 98 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide m ; III) G 0 or m. Therefore: SEI G m m whch mples SEI G m ; SEI G m. Moreover the equlty for the upper boud holds f d oly f codtos )-) hold d the equlty for the lower boud holds f d oly f codtos I)-III) hold. I order to prove the theorem we eed to fd m d m. Solvg equto for 0 we get 0.. Solvg for 0 we get cos Flly solvg equto for 0 we get 5 π 7 cos Now we wrte the tble of (ppromte) vlues of the fucto t zero pots d some rbtrry pots (oe smller th ll of them oe betwee ech two successve zero pots d oe lrger th ll of them). (see Tble ) From ths tble (Tble ) we c esly determe m d m. Completely logously t c be proved tht: Theorem 0. Let 0 d G be ucyclc chemcl grph wth vertces. It holds: f 0 ; f 5 π 7 SEI G cos ; 8 f 5 π 7 cos ; 8 Tble. Fucto vlues sgfct pots Crot. Chem. Act 8 (0) 9.

7 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide 99 SEI f 0 cos ; G f cos. Moreover the equlty the lower bouds holds f d oly f oe of the followg holds: ) 0 d G s cycle C ; b) d G oly hs vertces of degrees d ; c) d) e) 5 π 7 cos d G oly hs vertces of degrees d 8 ; 5 π 7 cos d G oly hs 8 vertces of degrees d ; 5 π 7 cos d G oly 8 hs vertces of degrees d ; c) the equlty the upper bouds holds f d oly f oe of the followg holds: ) 0 cos d G oly hs vertces of degrees d ; b) cos d G oly hs vertces of de- grees d ; cos d G s cycle C. Theorem. Let 0 d G be chemcl grph wth vertces. It holds: SEI G for wth equlty f d oly f G s -regulr grph; SEI G for wth equlty f d oly f ll vertces G hve degree or ; ) ) SEI G for wth equlty f d oly f G s -regulr grph; SEI G for wth equlty f d oly f ll vertces G hve degree or ; SEI G for wth equlty f d oly f G s -regulr grph; SEI G for wth equlty f G s P or C ; ) ) 5) 6) 7) SEI G for cos wth equlty f d oly f G s pth P ; cos for wth equlty f d oly f G s tree tht hs oly vertces of degrees d ; 9) SEI G for 0 cos wth equlty f d oly f G s tree tht hs oly vertces of degrees d. Proof: I order to prove )-6) t s suffcet to ote tht: ') m for ; ') m for ; ') m for ; ') m for ; 5') m for ; 6') m for. Now let us ssume tht. Let T be spg tree of G. Sce d the degrees of ll vertces T re ot greter th G (wth ll equltes f d oly f G T ) t follows tht the mmum s cheved f d oly f G T. Clms 7)-9) follow from the Theorem 8. 8) SEI G Theorem. Let 0 d G be chemcl grph Crot. Chem. Act 8 (0) 9.

8 00 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide wth vertces. It holds: SEI G for 0 d oly f G s -regulr grph; ) wth equlty f SEIG SEIG G G. SEI G for wth equlty f d oly f G oly hs vertces of degree d ; SEI G for wth equlty f d oly f G s tree whch ll vertces hve degrees d. ) ) Proof: I order to prove ) d ) t s suffcet to ote tht ') m for 0 ; for. Let us prove ). Suppose tht. If G s tree the clm follows from Theorem 8. Hece t s suffcet to prove tht ') m SEI G for ll cyclc chemcl grphs wth vertces. Suppose to the cotrry d let G be grph such tht SEI G G. Sce G s cyclc c 0. Let uv be the edge coted some cycle d let G be grph obted from G by elmtg the edge uv d by ddg oe pedt verte to ech of vertces u d v. Note tht: wth the smllest vlue of c m Sce cg cg O the other hd. SEI G SEI G t follows tht SEI G G. Ths s cotrdcto.. CONCLUSIONS I ths pper we hve lyzed etreml propertes of the vrble sum edeg de SEI G. We hve foud the grphs wth the etreml grphs the followg clsses of grphs (wth gve umber of vertces): ) clss of ll coected grphs ) clss of ll trees ) clss of ll uycyclc grphs ) clss of ll chemcl grphs 5) clss of ll chemcl trees 6) clss of ll chemcl uycyclc grphs 7) clss of ll grphs wth gve mml degree 8) clss of ll grphs wth gve mml degree 9) clss of ll trees wth gve umber of pedt vertces 0) clss of ll coected grphs wth gve umber of pedt vertces for ll SEI dces such tht. I the cse of 0 we hve restrcted our tteto to chemcl grphs chemcl trees d chemcl uycyclc grphs. We leve the soluto of the logous problem the remg seve clsses s ope problem. Further we propose solvg the logous set of problems for the two descrptors tht hve show good predctve propertes pper Ref.. Nmely we propose the study of the followg vrble descrptors: vrble verse sum deg de: d d \ 0 ; vrble verse sum lodeg de:. l d l d uve G uveg Further we propose the study of the geerlztos of dscrete Adrtc dces tht hve show good predctve propertes pper Ref.. The problem regrdg these geerlztos re etesos of the ope problems preseted pper Ref.. Nmely we propose the study of the followg vrble descrptors: vrble Rdć type lodeg de: l dul dv ; uveg Crot. Chem. Act 8 (0) 9.

9 D. Vukčevć Mthemtcl Propertes of the Vrble Sum Edeg Ide 0 vrble Rdć type d de: DD y \ 0 ; Rdć type ed de: \; uve G Du uve G Dv vrble sum lodeg de: l d l d d l d u u uve G vv G ; vrble verse sum lodeg de: ; l d l d uve G vrble verse sum deg: \ 0 ; uve G vrble msblce lodeg de: l du l dv ; uveg vrble msblce deg de: \ 0 ; vrble msblce edeg de: du dv \; uve G vrble msblce d de: \ 0 ; vrble m-m deg de: m du dv ; uve Gm du dv vrble m-m d de: m Du Dv ; m D D uv EG d d uveg uveg d D u u d v D v vrble m-m deg de: m du dv ; uve G m du dv vrble symmetrc dvso deg de: m du dv m du d v. uvegm du dv m du dv Ackowledgemets. The prtl support of Crot Mstry of Scece Educto d Sport (grts o d ) s grtefully ckowledged. Srh Mchele Rjtmjer s help cosderg Eglsh lguge s lso grtefully ckowledged. REFERENCES. D. Vukčevć d M. Gšperov Bod Addtve Modelg. Adrtc Idces Crot. Chem. Act 8 () (00) 60.. D. Vukčevć Bod Addtve Modelg. Mthemtcl propertes of M-m rodeg de Crot. Chem. Act 8 () (00) D. Vukčevć Bod Adtve Modelg. QSPR d QSAR studes of vrble Adrtc dces Crot. Chem. Act 8 () (00) I. Gutm B. Ruščć N. Trjstć d C. F. J. Wlco Jr Chem. Phys. 6 (975) S. Nkolć G. Kovčevć A. Mlčevć d N. Trjstć Crot. Chem. Act 76 (00). 6. R. Todesch d V. Coso Hdbook of Moleculr Descrptors Wley-VCH Wehem N. Trjstć Chemcl Grph Theory CRC Press Boc Rto P. Hse d D. Vukčevć Crot. Chem. Act 80 (007) D. Vukčevć d A. Grovc MATCH Commu. Mth. Comput. Chem. 57 (007) D. Vukčevć MATCH Commu. Mth. Comput. Chem. 57 (007) D. Vukčevć d A. Grovc MATCH Commu. Mth. Comput. Chem. 60 (008) 7.. H. Hu MATCH Commu. Mth. Comput. Chem. 60 (008) Crot. Chem. Act 8 (0) 9.