Hyper-wiener index of gear fan and gear wheel related graph

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1 Iteatoal Joual of Chemcal Studes 015; (5): 5-58 P-ISSN E-ISSN IJCS 015; (5): JEZS Receed: Accepted: We Gao School of Ifomato Scece ad Techology, Yua Nomal Uesty, Kumg , Cha. L Sh Isttute of Medcal Bology, Chese Academy of Medcal Sceces & Pekg Uo Medcal College, Kumg, 65009, Cha. Hype-wee dex of gea fa ad gea wheel elated gaph We Gao, L Sh Abstact The Hype-Wee dex, as a exteso of Wee dex, s a mpotat topologcal dex Chemsty. Thee s a ey close elato betwee the physcal, chemcal chaactestcs of may compouds ad the topologcal stuctue of that. The Hype-Wee dex s such a topologcal dex ad t has bee wdely used Chemsty felds. I ths pape, tems of peous studes, we deteme the Hype-Wee dex of gea fa gaph, gea wheel gaph ad the -cooa gaphs. Keywods: Chemcal gaph theoy, ogac molecules, Hype-Wee dex, fa gaph, wheel gaph, gea fa gaph, gea wheel gaph, -cooa gaph 1. Itoducto Chemcal compouds ad dugs ae ofte modeled as gaphs whee each etex epesets a atom of molecule, ad coalet bouds betwee atoms ae epeseted by edges betwee the coespodg etces. Ths gaph deed fom a chemcal compouds s ofte called ts molecula gaph, ad ca be dffeet stuctues. A dcato defed oe ths molecula gaph, the Wee dex, has bee show to be stogly coelated to aous chemcal popetes of the compouds. The Wee dex of a gaph s defed as the sum of dstaces betwee all pas of etces of the gaph. It has bee foud extese applcatos chemsty. Seeal yeas late, mathematca bega to pay atteto to the Wee dex ad study t fom the mathematcal pot of ew. I such backgoud, sce each stuctual featue of ogac molecule ca be expessed as a gaph, chemcal gaph theoy as a bach of combatoal chemsty s toduced to eseach the stuctue of molecule fom gaph theoy stadpot. The gaphs cosdeed ths pape ae smple ad coected. The etex ad edge sets of G ae deoted by V(G) ad E(G), espectely. The Wee dex s defed as the sum of dstaces betwee all uodeed pa of etces of a gaph G,.e., W ( G ) = d( u, ), { u, } V ( G) Coespodece: We Gao School of Ifomato Scece ad Techology, Yua Nomal Uesty, Kumg , Cha. Whee d( u, ) s the dstace betwee u ad G. Seeal papes cotbuted to deteme the Wee dex of specal gaphs. Gao ad Sh (Gao ad Sh, pess) detemed the Wee dex of gea fa gaph, gea wheel gaph ad the - cooa gaphs. Che (Che, 005) [] gaed the exact expesso fo geeal pepod gaph. Xg ad Ca (Xg ad Ca, 011) chaactezed the tee wth thd-mmum wee dex ad toduce the method of obtag the ode of the Wee dces amog all the tees wth ge ode ad damete, espectely. A tcyclc gaph s a coected gaph wth etces ad + edges. Wa ad Re (Wa ad Re, 01) [4] studed the Wee dex of tcyclc gaph whch hae at most a commo etex betwee ay two ccuts, ad the smallest, the secod-smallest Wee dces of the tcyclc gaphs ae ge. The Hype-Wee dex WW s oe of the ecetly dstace-based gaph aats. That WW clealy ecodes the compactess of a stuctue ad the WW of G s defe as: ~ 5 ~

2 Iteatoal Joual of Chemcal Studes WW ( G ) = 1 ( (, ) (, )) { u, d u } V ( G ) { u, d u } V ( G ) Pa (Pa, 01) deduced the fomula of Wee umbe ad Hype-Wee umbe of two types of polyomo systems. Moe esults o Wee dex ad Hype-Wee dex ca efe to [6-1]. The gaph F ={} P s called a fa gaph ad the gaph W ={} C s called a wheel gaph, whee P s a path wth etces ad C s a cycle wth etces. Gaph I (G) s called - cow gaph of G whch splcg hag edges fo eey etex G. The etex set of hag edges that splcg of etex s called -hag etces, ote *. By addg oe etex eey two adjacet etces of the fa path P of fa gaph F, the esultg gaph s a subdso gaph called gea F fa gaph, deote as. By addg oe etex eey two adjacet etces of the wheel cycle C of wheel gaph W, The esultg gaph s a subdso gaph, called gea wheel gaph, W deoted as. I ( ) I ths pape, we peset the Hype-Wee dex of F I ( ) ad W fst; the, the Hype-Wee dex of gea fa gaph ad gea wheel gaph ae detemed; at last, the Hype- Wee dex of -cooa gaph fo F ad W ae deed.. Ma esults ad poof Theoem 1. WW ( I( F )) = (5 4). 9 ( ) (6 1 ) 1 Poof. Let P = 1 ad the hagg etces of be,,, (1 ). Let be a etex F besde P, ad the 1 hagg etces of be,,,. By the defto of Hype-Wee dex, we hae I ths way, we get the decso. Theoem. WW ( I ( W )) = (5 ) 1 5 (6 1) ( ). 1 Poof. Let C = 1 ad,,, be the hagg etces of (1 ). Let be a etex W besde C, ad 1,,, be the hagg etces of. By the defto of Hype-Wee dex, we hae ~ 5 ~

3 Iteatoal Joual of Chemcal Studes Thus, the dese esult s ge. Theoem 4. WW ( W 5 9 ) = Poof. Let C = 1 ad be a etex W besde C. Let, 1 be the addg etex betwee ad +1. I ew of the defto of Hype-Wee dex, we deduce Hece, we dee the dese cocluso. Takg =0 Theoem 1 ad Theoem, we get followg two coollaes. Coollay 1. WW ( F ) = Coollay. WW ( W ) = 9 Now, let us beg dscussg the gea elated gaphs. Theoem. WW ( F ) = , 1 Poof. Let P=1 ad be the addg etex betwee ad +1. Let be a etex F besde P. By tue of the defto of Hype-Wee dex, we get ~ 54 ~

4 Iteatoal Joual of Chemcal Studes Hece, we get the dese cocluso. Poof. Let P = 1, 1 ad be the addg etex betwee 1 ad +1. Let,,, be the hagg etces of 1 (1 ). Let, 1, 1,, 1,, be the hagg etces (1-1). Let be a etex F besde P, ad the, 1 of hagg etces of be 1,,,. By tue of the defto of Hype-Wee dex, we get ~ 55 ~

5 Iteatoal Joual of Chemcal Studes Thus, the esult s hold. Theoem 6. WW ( I ( W 61 5 )) = ( ) 15 9 ( ) ( ). Poof. Let C = 1 ad be a etex W besde C., 1 1 be the addg etex betwee ad +1. Let,, 1, be the hagg etces of ad,,, be the 1 hagg etces of (1 ). Let, 1 = 1,, 1 ad,, 1, 1,, be the hagg etces of, 1 (1 ). I ew of the defto of Hype-Wee dex, we deduce ~ 56 ~

6 Iteatoal Joual of Chemcal Studes As cocluso, we obta the fal cocluso.. Cocluso Combatoal chemsty s a ew poweful techology molecula ecogto ad dug desg. It s a wet-laboatoy methodology puposed to massely paallel sceeg of chemcal compouds fo the foudg of compouds that hae ceta bologcal acttes. The powe of tck daws fom the teacto betwee computatoal modelg ad expemetal desg. Fa gaph, wheel gaph, gea fa gaph, gea wheel gaph ad the -cooa gaphs ae commo stuctual featues of ogac molecules. The cotbutos of ou pape ae detemg the Hype-Wee dex of these specal stuctual featues of ogac molecules. 4. Ackowledgemets Fst, we thak the eewes fo the costucte commets mpog the qualty of ths pape. Ths wok was suppoted pat by the PHD stat fudg of fst autho. We also would lke to thak the aoymous efeees fo podg us wth costucte commets ad suggestos. 5. Refeeces 1. Gao W, Sh L. Wee dex of gea fa gaph ad gea wheel gaph. Asa Joual of Chemsty, pess.. Che DQ, The alue of the Wee dex of the peptods. J Wuha Ist Chem Tech 005; 7(): Xg BH, Ca GX. The Wee dex of tees wth pescbed damete. Opeatos Reseach Tasactos, 011; 15(4): Wa H, Re HZ. The Wee dex of a class of tcyclc gaphs. Joual of Mathematcal Study 01; 45(): Pa YJ. Wee umbe ad hype-wee umbe of two types of polyomo systems. Joual of Mathematcal Study 01; 46(): Steaoc D. Maxmzg Wee dex of gaphs wth fxed maxmum degee. Match Commu Math Comput. Chem 008; 60:71-8. ~ 57 ~

7 Iteatoal Joual of Chemcal Studes 7. Hog Y, Lu HQ, Wu XY. O the Wee dex of ucyclc gaphs. Hacettepe Joual of Mathematcs ad Statstcs 011; 40(1): Zhag XD, Xag QY. The Wee dex of tees wth ge degee sequeces. Match Commu. Math. Comput. Chem 008; 60: Cash G. Thee methods fo calculato of the Hypewee dex of molecula gaphs. J Chem If Comput Sc 00; 4, Casto E, Gutma I, Mao D, Peuzzo, P. Upgadg the wee dex. J Seb Chem Soc 00; 67(10): Wu B. Wee dex of le gaphs. Match Commu Math Comput Chem 010; 64: Vjayabaath A, Ajaeyulu GSGN. Wee dex of a gaph ad chemcal applcatos. Iteatoal Joual of ChemTech Reseach 01; 5(4): Dakelma P, Gutma I, Mukwemb S, Swat HC. The edge-wee dex of a gaph. Dscete Mathematcs, 009; 09:4, ~ 58 ~