Some Lower and Upper Bounds on the Third ABC Co-index

Transcription

1 Intenational JMath Combin Vol4(07), Some Lowe an Uppe Bouns on the Thi BC Co-inex Deepak S Revanka, Piyanka S Hane, Satish P Hane 3 an Vijay Teli 3 Depatment of Mathematics, KLE, D M S S C E T, Belagavi , Inia Depatment of Mathematics, KLS, Gogte Institute of Technology, Belagavi , Inia 3 Depatment of Mathematics,KLS, Vishwanathao Deshpane Rual, Institute of Technology, Haliyal , Inia evankas@gmailcom, piyanka8hane@gmailcom, hanesp33@gmailcom, vijayteli@gmailcom bstact: Gaonac efine the secon BC inex as BC (G) = + n i n j n in j v i v j E(G) Dae Won Lee efine the thi BC inex as BC 3(G) = v i v j E(G) + e i e j e ie j an stuie lowe an uppe bouns In this pape, we efine a new inex which is calle thi BC Coinex an it is efine as BC 3(G) = v i v j / E(G) + n i n j n in j an we foun some lowe an uppe bouns on BC 3(G) inex Key Wos: Molecula gaph, the thi atom - bon connectivity (BC 3) inex, the thi atom - bon connectivity co-inex (BC 3) MS(00): 05C40, 05C99 Intouction The topological inices plays vital ole in chemisty, phamacology etc [] Let G = (V, E) be a simple connecte gaph with vetex set V (G) = {v, v,, v n } an the ege set E(G), with V (G) = n an E(G) = m Let u, v V (G) then the istance between u an v is enote by (u, v) an is efine as the length of the shotest path in G connecting u an v The eccenticity of a vetex v i V (G) is the lagest istance between v i an any othe vetex v j of G The iamete (G) of G is the maximum eccenticity of G an aius (G) of G is the minimum eccenticity of G Receive May 7, 07, ccepte Novembe 9, 07

2 Some Lowe an Uppe Bouns on the Thi BC Co-inex 85 The Zageb inices have been intouce by Gutman an Tinajstic []-[5] They ae efine as, M (G) = i, M (G) = i j v i V (G) v iv j V (G) The Zageb co-inices have been intouce by Doslic [6], M (G) = ( i + j ), M (G) = Similaly Zageb eccenticity inices ae efine as E (G) = v i V (G) e i, E (G) = v iv j V (G) Estaa et al efine atom bon connectivity inex [7-0] BC(G) = an Gaovac efine secon BC inex as BC (G) = v iv j E(G) v iv j E(G) + i j i j e i e j +, n i n j n i n j ( i j ) which was given by eplacing i, j to n i, n j whee n i is the numbe of vetices of G whose istance to the vetex v i is smalle than the istance to the vetex v j [-4] Dae an Wan Lee efine the thi BC inex [6] BC 3 (G) = v iv j E(G) + In this pape, we have efine the thi BC co - inex; BC 3 (G) as BC 3 (G) = foun some lowe an uppe bouns on BC 3 (G) + Lowe an Uppe Bouns on BC 3 (G) Inex Calculation shows clealy that (i) BC 3 (K n ) = 0; (ii) BC 3 (K,n ) = ( ) n ;

3 86 Deepak S Revanka, Piyanka S Hane, Satish P Hane an Vijay Teli (iii) BC 3 (C n ) = (n 3) n ; 4n (iv) BC 3 (C n+ ) = n(n 3) (n ) Theoem Let G be a simple connecte gaph Then BC 3 (G) E (G), whee E (G) is the secon zageb eccenticity coinex Poof Since G K n, it is easy to see that fo evey e = v i v j in E(G), e i + e j 3 By the efinition of BC 3 coinex BC 3 (G) = + ei e j = ¾ e i e j E (G) Theoem Let G be a connecte gaph with m eges, aius = (G), iamete = (G) Then, m m BC3 (G) with equality hols if an only if G is self-centee gaph Poof Fo e i, e j, + + ( ) (as e j, 0 = e i e j e j e i + ( e j )) + ( ) (as e i an ( ) 0) + ( ) with equality hols if an only if e i = e j = Similaly we can easily show that, + ( ) fo e i, e j with equality holing if an only if e i = e j = ¾ The following lemma can be veifie easily Lemma Let (a, a,, a n ) be a positive n-tuple such that thee exist positive numbes

4 Some Lowe an Uppe Bouns on the Thi BC Co-inex 87, a satisfying 0 < a a i Then, n n a i i= ( n a i ) 4 ( i= a + a ) a with equality hols if an only if a = o q = ( a) + n is an intege an q of numbes a i coincie with a an the emaining n q of the a is coincie with ( a) Theoem 3 Let G be a simple connecte gaph with m eges, = (G), = (G) Then, BC 3 (G) = 4m ( )( ) ( + ) E(G) Poof By Theoem we know that +, vi v j / E(G) () lso by Lemma 3 we have a a i () Let a = an ai = +, v i v j / E(G) an = in equations () an () We theefoe know that a i i= n n ( a i ) 4 ( a + ) a, ie, which implies that ( a i ) n a i 4( a + ), a ( ) 4n a ai i ( a + ) 4n a i [ a a + ] 4n a i [ ] 4n a i a a +a [ + a] an

5 88 Deepak S Revanka, Piyanka S Hane, Satish P Hane an Vijay Teli an v iv j E(G) + 4n [ 8n [ ] + ] + ( + ) ( e i + e j e i e j ) Theefoe 8n ( )( ) ( [ ] + ) 8n = ( )( ) [ e i e j e i e ] j v iv j + We know that fom Theoem Thus, + ( + ) E (G) + 4m ( )( ) ( )E + (G), ( + ) 4m ( )( ) ( + ( + ) ) E (G) Theoem 4 Let G be a simple connecte gaph with n vetices an m eges Then, BC 3 (G) nm nm (G) m n m nm (G) + M (G) ¾ Poof Fom Theoem, Since e i (n i ), we know that + e i e j e i e j This completes the lowe boun (n i )(n j ) = = (n n i n j + i j ) mn nm (G) + M (G)

6 Some Lowe an Uppe Bouns on the Thi BC Co-inex 89 Now, since G K n, e i e j fo v i v j / E(G), we get that + By Cachy-Schwaz inequality, we also know that ei + e j Since e i n i fo v i V (G), we get that m m (e i + e j ) (n i + n j ) n = [ ] m nm M (G) m ei + e j ( i + j ) (e i + e j ) = m n mm (G) m ¾ Refeences [] R Toeschini, V Consonni, Hanbook of Molecula Desciptos, Wiley - VCH Weinheim, 000 [] K C Das, Maximizing the sum of the squaes of the egees of a gaph, Discete Math, (004), [3] K C Das an I Gutman, Some popeties of secon Zageb inex, MTCH Commun Math Comput Chem, 5,(004), 03 - [4] K C Das, I Gutman an B Zhou, New uppe bouns on Zageb inices, JMath Chem, 56,(009), 54-5 [5] I Gutman an K C Das, The fist Zageb inices 30 yeas afte, MTCH Commun Math Comput Chem, 50,(004), 83-9 [6] T Dovslic, Vetex weighteweine polynomial fo composite gaphs, s Math Contemp,,(008), [7] KC Das, DW Lee an Gaovac, Some popeties of the Zageb eccenticity inices, s Math Contemp, 6 (03), 7-5 [8] DW Lee, Lowe an uppe bouns of Zageb eccenticity inices on unicyclic gaphs, v ppl Math Sci, (7) (03),

7 90 Deepak S Revanka, Piyanka S Hane, Satish P Hane an Vijay Teli [9] M Ghobani, M Hosseinzaeh, new vesion of Zageb inices, Filomat, 6 (0), [0] D Vukicevic, Gaovac, Note on the compaison of the fist an secon nomalize Zageb eccenticity inices, cta Chim Slov, 57 (00), [] E Estaa, L Toes, L Roguez, I Gutman, n atom-bon connectivity inex: Moelling the enthalpy of fomation of alkanes, Inian J Chem, 37 (998), [] E Estaa, tom-bon connectivity an the enegetic of banche alkanes, Chem Phys Lett, 463 (008), 4-45 [3] KC Das, tom-bon connectivity inex of gaphs, Discete ppl Math, 58 (00), 8-88 [4] KC Das, I Gutman, B Futula, On atom-bon connectivity inex, Chem Phys Lett, 5 (0), [5] Dae Won Lee, Some lowe an uppe bouns on the thi BC inex, KCE Intenational Jounal of Gaphs an Combinatoics, 3, (06), - 5 [6] D S Revanka, S P Hane, S R Jog, P S Hane an M M Patil, Zageb coinices on some gaph opeatos of Tapole gaph, Intenational Jounal of Compute an Mathemaical Sciences, 3 (3), (04)