Steie Hype Wiee Idex A. Babu 1, J. Baska Babujee Depatmet of mathematics, Aa Uivesity MIT Campus, Cheai-44, Idia. Abstact Fo a coected gaph G Hype Wiee Idex is defied as WW G = 1 {u,v} V(G) d u, v + d u, v is the extesio of wiee idex. Chatad et al. itoduced Steie wiee idex i 1989 is a geealizatio of the cocept of gaph distace. Fo a coected gaph G of ode ad S V (G), the Steie distace d(s) of the vetices of S is the miimum size of a coected subgaph whose vetex set is S. Li, Mao, ad Gutma geealized the cocept of wiee idex of a gaph G as the Steie wiee idex deoted as SW (G) = d(s). I this pape we itoduce the Steie Hype Wiee Idex SWW (G) ad study some stadad gaph stuctues as well as some popeties ad bouds fo it. Keywods Gaphs, Degee, Distace, Topological idices, steie distace. I. INTRODUCTION Let G be a simple coected gaph, whose vetex ad edge sets ae deoted as V (G) ad E(G) espectively, ad V (G) =, E(G) = m called ode ad size of the gaph G. The degee deg G (v) of a gaph G is cadiality of the fist eighbos of the vetex u ad x, y V (G) the the distace d(x, y) = d G (x, y) is the shotest path betwee u ad v. Fo a coected gaph G, The Wiee idex [14] W(G) of the gaph G is deoted as W(G) = d G (x, y) {x,y} V(G) ad the Hype Wiee idex WW(G) of a simple coected gaph G is deoted as WW(G) = d G x, y + d G x, y {x,y} V(G) d G (x, y) is the distace betwee x ad y. Wiee idex ad Hype Wiee idex WW(G) ae studied fo may types of chemical gaphs[7]. I 1989 Chatad et al itoduced the cocept of Steie distace of a coected gaph [] is a geealizatio of the aciet gaph distace. Fo a coected gaph G of ode at least ad S V (G), the Steie distace d(s) of the vetices of S is the miimum size of a coected subgaph whose vetex set is S. I view of equatio (1) Li, Mao, ad Gutma geealized the cocept of wiee idex of a gaph G as the Steie wiee idex [1] deoted as SW (G) = d(s) Whe S = x, y, S =, the the stiee distace educes to distace betwee a pai of vetices which is equal to the odiay wiee idex [14] that is W(G) = SW (G) = d(s) S = Futhe whe = 0, SW (G) = 0, ad = 1, SW G = 1. Klei et al itoduced the cocept of hype wiee idex [4]. The Hype Wiee Idex has bee extesively studied because of widespead use i ISSN: 31-5373 http://www.ijmttjoual.og Page 8
chemical coelatio. I this pape, we itoduce Steie hype Wiee Idex ad study some iteestig popeties ad bouds. II. STEINER HYPER WIENER INDEX OF STANDARD GRAPH STRUCTURES Steie Hype Wiee Idex of a simple coected gaph G is the geealizatio of Hype Wiee Idex with vetices. I view of equatio (1) ad (3), we itoduce the followig defiitio Defiitio.1. Fo ay coected gaph G the Steie Hype Wiee idex SWW (G) of a gaph G is defied as SWW (G) = [d S + d S ] Whee 1 1 ad whe = 1 the SWW (G) = 0. Oe ca ote that ithe special case = of equatio (5) implies Hype Wiee idex. Defiitio.. Fo a coected gaph G the steie Hype wiee polyomial of a gaph G is defied as SWW G, x = x d(s) S= d s + x the fist deivative of the Steie Hype wiee polyomial withx = 1 gives the SWW (G). Theoem.3. The Steie Hype Wiee idex of the Sta gaph S is whee. SWW (S ) = 1 [ + 1 ] Poof. Let v 1 be the cete vetex of the sta gaph S. Divide the vetex set V (G) of S i to two patitio as follows. Fo ay S V (S) ad S =, if v 1 S, the d S (S) = ad v S deg S v =. If v 1 S, the d S S = ad v S deg S v = 1 Theefoe SWW (S ) = [d S + d S ] + [d S + d S ] = 1 v 1 S, v 1 S, [ + + 1 = 1 = 1 + + 1 1 [ + 1 ] k + k Popositio.4. Fo a complete gaph K with vetices ad be a itege the SWW (K) = ( 1)[1 + ] Equivaletly oe ca itepet the Steie Hype Wie Idex ISSN: 31-5373 http://www.ijmttjoual.og Page 9
SWW (K ) = SW (K ) + 1 Theoem.5. The Steie Hype Wiee idex of path P of ode is SWW P = SW P + d S S V(G) whee, ad SW (P ) is the steie Weie Idex of the path P. Poof. Let V P = {v 1, v,, v } be the vetices of P whee v 1 ad v ae pedetvetices. Fo ay S V (P) ad S = we have v 1 &v S,v 1 o v S,ad v 1 &v S. Hece Hece SWW P = [d S + d S ] + [d S + d S ] + [d S + d S ] S V P v 1 S o v S S V P v 1 ad v S = d(s) + d S S V(G) = SW G + d S S V P v 1 ad v S Theoem.6. Let G be the K m, be the complete bipatite gaph with m + vetices, ad beig a itege such that m +, the = [ m + m + + if 1 m m + + if m < m + if < m + Poof. Let G = K m, ad let V 1 = {x 1, x, x 3,, x m } ad V = {y 1, y, y 3,, y } be the two patitio of vetices of G. Case I. 1 m Fo all S V (G) ad S =, we have the followig thee subcases (i) S V1 = (ii) S V = (iii) S V 1 = ad S V =. If S V 1 = o S V = the S _ V suppose S = {y 1, y,, y }. The the Steie tee cotaiig the vetices y 1, y,, y has edges theefoe d G (S) =. Similaly, if S V = the d G S = ad suppose S V 1 = ad S V = ad let S = {x 1, x,, x a y 1, y,, y ( m ) the the Steie tee iduced by the edgesx 1 y 1, y 1 x, y 1 x 3,, y 1 x a, x 1 y, x 1 y 3,, x 1 y m } Theefoe d G S = Thus SWW K m, = [d G S + d G S ] + [d G S + d G S ] + [d G S + d G S ] S V K m, S V 1 = S V K m, S V = SWW K m, = SWW K m, + m + + m + S V K m, S V 1 = V m ISSN: 31-5373 http://www.ijmttjoual.og Page 10
= SWW K m, + m + + m + Case II: Coside m <. Fo ay S V (G) ad S =, we have S V 1 = o S V 1 =. If S V 1 = the S V ad suppose S = {y 1, y,, y } the the tee T iduced by the edges is {x 1 y 1, x 1 y,, x 1 y m } is a Steie tee cotaiig S hece d G (S) =. If S V 1 ad let S = x 1, x,, x a, y 1, y, y m (a ) the the tee iduced by the vetices S has the edges {x 1 y 1, y 1 x, y 1 x,, y 1 x m, x 1 y, x 1 y 3,, x 1 y m } is a steie tee cotaiig S. Theefoe d G S =. SWW K m, = 1 = = + 1 1 + + 1 1 + m + + 1 + 1 Case III: we coside the emaiig case < m +. Fo ay set S V(G) with vetices. If S V 1, ad S V suppose S = {x 1, x,..., x x, y 1, y,..., y x }. The the steie tee T iduced by the edges is {x 1 y 1, y 1 x,..., y 1 x x, x 1 y, x 1 y 3,..., x 1 y x Theefoe d G (S) = Thus m + SWW (K m, ) = () + m + = () m + Remaks.7 Fo a coected gaph G with vetices ad m edges the SWW (G) = ( 1). Theoem.8 let T be a tee with vetices ad havig p pedet vetices Thus SWW 1 (T) = p( )( 1) + ( 1)( p). Poof. Fo = 1 we have the followig two cases. Let v be the pedet vetices such that v V(G) \ S is pedet, The the vetices cotaied i S fom a tee of ode 1 Theefoe d T (S) = ad deg T (v) v S = m 1. Thee ae such subsets with cadiality 1 i V(G) \ S. If V G \S is o pedet i S. The the vetices cotaied i S caot fom a tee. The the espective Steie tee must cotai all the vetices of T. Theefoe d T (S) = 1 ad Thee ae p such subsets. Hece u S deg T (u) = m deg T (u) whee v V(G)\S. SWW 1 (T) = d G S + d G S + d G S + d G S S V T v V G \S S V T v S = p[( ) + ] + ( p)[( 1) + 1 ] = p( )( 1) + ( 1)( p) III. SOME BOUNDS FOR STEINER RECIPROCAL DEGREE DISTANCE INDEX Fo a coected Gaph G the geatest ad smallest vetex degee of the gaph G espectively deote by (G) ad δ(g). The followig Popositio, follows immediately fom the defiitios of the Steie Hype Wiee Idex, equatio (). Obsevatio 3.1. Let G be a coected gaph with vetices ad let T be the spaig tee ISSN: 31-5373 http://www.ijmttjoual.og Page 11
SWW (G) SWW (T) holds fo all, with equality holds iff G is a tee. Fo a coected Tee T theoem 3.3 of 3 we have followig bouds Poof. We have 1 1 SW G + 1 + 1 SWW G = d S + d S S V(G) (8) Fom the above lemma 8 = SW G + d S 1 ( 1) SW (G) + 1 + 1 1 1 + d S () Theoem 3.: Let T be the tee with vetices ad let ( ) the Fom the defiitio of SW k (G) we have Ad Hece + SWW (G) + 1 + 1 () SW (G) 1 SWW (G) 1 1 () SWW (G) ( 1) + 1 1 IV. CONCLUSIONS The Steie Hype Weie idex itoduced i this pape will have applicatio i the study of QSAR (Quatitative Stuctue-Popety Relatioship) ad QSPR (Quatitative Stuctue-Popety Relatioship) study sice it is a combiatio of Steie distace ad Hype Weie idex. It is easy to fid the Steie Steie Hype Weie idex of wheel gaph, widmill gaph, catepilla, ad Catesia poduct of stadad gaphs. Ivestigatig the geeal gaph is ou futue wok. REFERENCES [1] V. Adova, D. Dimitov, J. Fik, R. Skekovski, Bouds o Gutma idex, MATCHCommu. Math. Comput. Chem, J. Algeba, vol. 67 pp. 515-54, 01. [] G. Chatad, O.R. Oellema, S. Tia ad H.B. Zou, Steie distace i gaphs, Casopis Pest. Mat., vol. 114, 399-410, 1989. [3] A.A. Dobyi, R. Etige, I. Gutma, Wiee idex of tees theoy ad applicatios. Acta Appl [4] B. Futula, I. Gutma, H. Li, Moe tees with all degees havig extemal Wieeidex, MATCH Commu. Math. Comput. Chem. Vol. 70, 93-96, 013. [5] I. Gutma, Selected popeties of the Schultz molecula topological idex, J. Chem.If. Comput. Sci., vol. 34, 1087-1089, 1994. [6] H. Hosoya, O some coutig polyomials i chemisty, Disc. Appl. Math., vol. 1939-57, 1988. ISSN: 31-5373 http://www.ijmttjoual.og Page 1
[7] Hogbo Hua, Sheggui Zhag, O the ecipocal degee distace of gaphs, DisceteApplied Mathematics., vol. 160, 115-1163, 01. [8] Kexiag Xu, Kika Ch. Das, O Haay idex of gaphs,discete Applied Mathematics., vol.159, 1631-1640, 011. [9] Kog-Mig Chog, The Aithmetic Mea-Geometic Mea Iequality, A New Poof,Mathematics Magazie., vol. 49,87-88, 1976. [10] D. Plavi, S. Nikoli, N. Tiajsti, Z. Mihali, O the Haay idex fo the chaacteizatio of chemical gaphs, J. Math. Chem., vol.1 35-50, 1993. [11] Yapig Mao et.al, Steie Wiee idex of Gaph Poducts, Tasactios o Combiatoics., vol. 5, 39-50, 016. [1] Xueliag Li et.al., The Steie Wiee Idex of a Gaph, Discussioes MathematicaeGaph Theoy,vol. 36, 455-465, 016. [13] Yapig Mao et.al, Steie Degee Distace, MATCH Commu. Math. Comput. Chem., vol. 78, 1-30, 017. [14] H. Wiee, Stuctual detemiatio of paaffi, boilig poits, J. Am. Chem.Soc., vol. 69 17-0, 1947. [15] Yapig Mao, Steie Distace i Gaphs-A Suvey, axiv:1708.05779v1[math.co] 18Aug 017. [16] Yapig Mao, Steie Haay Idex,Kagujevac Joual of Mathematics., vol. 4(1),9-39,018. ISSN: 31-5373 http://www.ijmttjoual.og Page 13