MATCH Counications in Mathatical and in Coputr Chistry MATCH Coun. Math. Coput. Ch. 79 (2018) 477-489 ISSN 0340-6253 So Inqualitis for Gnral Su Connctivity Indx I. Ž. Milovanović, E. I. Milovanović, M. Matjić Faculty of Elctronic Enginring, Bogradska 14, P.O.Box 73, 18000 Niš, Srbia {igor, a}@lfak.ni.ac.rs (Rcivd Fbruary 17, 2017) Abstract Lt G b a sipl connctd graph with n vrtics and dgs. Dnot by d 1 d 2 d n > 0andd( 1 ) d( 2 ) d( ) > 0 squncs of vrtx and dg dgrs, rspctivly. Adjacncy of th vrtics i and j is dnotd by i j. A vrtx-dgr topological indx, rfrrd to as gnral su-connctivity indx, is dfind as χ α = χ α (G) = i j (d i + d j ) α,whrαis an arbitrary ral nubr. Lowr and uppr bounds for χ α ar obtaind. W also prov on gnralization of discrt Kantorovich inquality. 1 Introduction Lt G =(V,E), V = {1, 2,...,n}, E = { 1, 2,..., } b a sipl connctd graph with n vrtics and dgs. Dnot by d 1 d 2 d n > 0andd( 1 ) d( 2 ) d( ) > 0 squncs of vrtx and dg dgrs, rspctivly. If vrtics i and j ar adjacnt, w dnot it as i j. In addition, w us th following notation: Δ = d 1, δ = d n,δ = d( 1 )+2,δ = d( ) + 2. As usual, L(G) dnots a lin graph of G. Gutan and Trinajstić [1] introducd two vrtx dgr topological indics, nad as th first and th scond Zagrb indx. Ths ar dfind as M 1 = M 1 (G) = n d 2 i and M 2 = M 2 (G) = d i d j. i j
-478- Th first Zagrb indx can b also xprssd as (s [23]) M 1 = M 1 (G) = i j (d i + d j ). (1) Dtails on th athatical thory of Zagrb indics can b found in [2 6, 21, 24]. Rcntly [7], a graph invariant siilar to M 1 ca into th focus of attntion, dfind as n F = F (G) = d 3 i, which for historical rasons [3] was nad forgottn topological indx. It satisfis th idntitis F = (d 2 i + d 2 j)= [d( i )+2] 2 2M 2. (2) i j Anothr dgr basd graph invariant was introducd in [8], and nad gnral suconnctivity indx, χ α. It is dfind as χ α = χ α (G) = (d i + d j ) α, (3) i j whr α is an arbitrary ral nubr. Mor on athatical proprtis of this indx can b found in [9 14]. In this papr w ar concrnd with uppr and lowr bounds for χ α. Also, w prsnt on gnralization of discrt Kantorovich inquality, and show how it can b usd to obtain uppr bounds for M 1. Th drivd inquality is bst possibl in its class. 2 Prliinaris In this sction w rcall so rsults for χ α, and stat a fw analytical inqualitis ndd for our work. In [10] (s also [9]) th following was provd: La 1. [10]. Lt G b a nontrivial connctd graph with axiu dgr Δ and iniu dgr δ, and α R. Thn 2 α 1 Δ α 1 M 1 χ α 2 α 1 δ α 1 M 1, if α<1, (4) 2 α 1 δ α 1 M 1 χ α 2 α 1 Δ α 1 M 1, if α 1. (5) Th quality holds in ach inquality for so α 1if and only if G is rgular.
-479- In [8] uppr and lowr bounds for χ α (G) in trs of invariant M 1 and graph paratr wr obtaind. La 2. [8]. LtG b a graph with 1 dgs. If 0 <α<1, thn χ α (G) M1 α 1 α, (6) and if α<0 or α>1, thn χ α (G) M α 1 1 α. (7) Equality holds if and only if d i + d j is constant, for any dg {i, j} E. For th ral nubr squncs th following rsult was provd in [15] (s also [16]): La 3. [15]. Lt p =( ), and a =(a i ), i =1, 2,...,, b two positiv ral nubr squncs with th proprtis =1 and 0 <r a i R<+. Thn a i + rr r + R, (8) a i with quality if and only if for so k, 1 k, holds R = a 1 = = a k a k+1 = = a = r. In [19] th following was provd: La 4. [19]. Lt q =(q i ) b a squnc of positiv ral nubrs, and a =(a i ) and b =(b i ) squncs of ral nubrs with th proprtis 0 <r 1 a i R 1 < + and 0 <r 2 b i R 2 < +, i = 1, 2,...,. Dnot with S a subst of I = {1, 2,...,} which iniizs th xprssion Thn q i q i a i b i q i a i q i 1 q i. 2 i S q i b i (R 1 r 1 )(R 2 r 2 ) i S ( q i q i ) q i. (9) i S
-480- In th following la w rcall wll-known Chbyshv inquality (s for xapl [16]) which will b usd latr. La 5. Lt q =(q i ) b a squnc of positiv ral nubrs, and a =(a i ) and b =(b i ), i =1, 2,...,, squncs of non-ngativ ral nubrs of siilar onotonicity. Thn q i q i a i b i q i a i q i b i. (10) If squncs a =(a i ) and b =(b i ) has opposit onotonicity, thn th sns of (10) rvrss. 3 Main rsult 3.1 A nw inquality for ral nubr squncs In this sction w prov a nw inquality for ral nubr squncs. Thor 1. Lt p = ( ) and a = (a i ), i = 1, 2,...,, b ral nubr squncs, with a = (a i ) bing onotonic and 0 < r a i R < +. Lt S b a subst of I = {1, 2,...,} which iniizs th xprssion 1. 2 Thn whr a i a i γ(s) = i S (1+γ(S) i S ) ( ) 2 (R r)2, (11) rr 1 i S. Equality is attaind if R = a 1 = = a = r. Proof. For q i = p, a i = a i, b i = 1 a i, R 1 = R, r 1 = r, R 2 = 1 and r r 2 = 1, R i i =1, 2,...,, th inquality (9) bcos a i a i ( 1 1 ( ) 2 (R r) r 1 ) i S R 1 i S. (12) p i
-481- For q i = p, a i = a i, b i = 1 a i, i =1, 2,...,, th inquality (10) transfors into i 1 a i a i ( ) 2. (13) Cobining (12) and (13), givs a i a i (R r) 2 i S ( ) 2 1+ rr whrfro w arriv at (11). 1 i S Rark 1. Th inquality (11) is a rvision of th inquality a i a i ( 2, R + +1 )( 2 r +1 2 R + ) 2 r rr 2 givn in [17]. Th abov inquality is not always corrct. It is corrct whn = 1, i =1, 2...,. Howvr, if 1 and p 1 + p 2 + + p =1, th abov inquality ight b incorrct. Thus, for xapl for =5, p 1 = p 2 = 1, p 4 3 = p 4 = p 5 = 1, a 6 1 = a 2 =3, a 3 = a 4 = a 5 =2, r =2and R =3, on obtains that 625 624, which is obviously wrong. Sinc γ(s) 1 for ach S I 4, th following corollary of Thor 1 is valid. Corollary 1. Lt p =( ), b a squnc of positiv ral nubrs and a =(a i ), i = 1, 2,...,, a onoton squnc of positiv ral nubrs, with th proprtis p 1 + + p =1, 0 <r a i R<+. Thn a i a i (R + r)2 4rR. (14) Rark 2. Th inquality (14) (provd in [20]) is a gnralization of Kantorovich inquality (s for xapl [16]). For =1,i =1, 2,...,, th following corollary of Thor 1 holds:
-482- Corollary 2. Lt a =(a i ), i =1, 2,...,, b a ral nubr squnc with th proprty 0 <r a i R<+. Thn ) 1 a i 2 (R r)2 (1+α(), (15) a i rr whr α() = 1 ( ) 1 ( 1)+1 +1. 4 2 2 Rark 3. Th inquality (15) was provd in [17]. Sinc α() 1, it is a gnralization 4 of th inquality provd in [22]. a i 1 2 a i 4 (R + r)2 rr, 3.2 So inqualitis for gnral su connctivity indx In what follows w driv lowr and uppr bounds for th dgr-basd topological indx χ α in trs of topological indics M 1, M 2 and F and graph paratrs, Δ and δ. Thor 2. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Thn, for any α 2, If α 1, thn If α 0, thn (F +2M 2 )δ α 2 M 1 δ α 1 χ α (F +2M 2 )Δ α 2. (16) χ α M 1 Δ α 1. (17) δ α χ α Δ α. Equalitis in th abov inqualitis ar attaind, rspctivly, for α =2, α =1, α =0,or if L(G) is rgular. Whn α 2, α 1 and α 0, rspctivly, th opposit inqualitis ar valid. Proof. Lt = {i, j} b an arbitrary dg of graph G. Thnd() =d i +d j 2. According to (3), topological indx χ α can b coputd fro th following xprssion χ α = (d i + d j ) α = (d( i )+2) α, χ 0 =. (18) i j Fro (3) follows F +2M 2 = χ 2 = (d i + d j ) 2 = (d( i )+2) 2. i j
whr EF is th rforulatd forgottn topological indx. Whn α 3, th opposit inquality is valid. Obviously, ths inqualitis dpnd on a larg nubr of graph invariants. Anothr qustion is how would (19) look lik if β 4 and its practical usability. For α = 1 and β = 1, th inquality (19) givs a connction btwn haronic and 2 su-connctivity indics. -483- Sinc for α 2 holds i.. χ α = (d( i )+2) α = (d( i )+2) 2 (d( i )+2) α 2, δ α 2 (d( i )+2) 2 χ α Δ α 2 (F +2M 2 )δ α 2 (d( i )+2) 2, χ α (F +2M 2 )Δ α 2. By a siilar procdur, th raining inqualitis in Thor 2 can b provd. Rark 4. Lt α and β b arbitrary ral nubrs such that α β 0. Thn, according to χ α = (d( i )+2) α = (d( i )+2) β (d( i )+2) α β follows that δ α β χ β χ α Δ α β χ β, (19) with quality if and only if α = β, orl(g) is rgular. If α β 0, th opposit inquality is valid. Th qustion is for which valus of paratr β th inquality (19) has practical iportanc. For β =0, β =1and β =2it was considrd in Thor 2. Sinc (d( i )+2) 3 = EF +6F +12M 2 12M 1 +8, for α 3 holds δ α 3 (EF +6F +12M 2 12M 1 +8) χ α Δ α 3 (EF +6F +12M 2 12M 1 +8), Rark 5. Sinc 2δ δ Δ 2Δ, thn for α 1 and α 1, fro(17) th inqualitis (4) and (5) ar obtaind. Hnc, th inquality (17) is strongr than ths inqualitis.
-484- Corollary 3. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Thn, for any α 2 Equality is attaind if G is rgular. 4M 2 δ α 2 χ α 2F Δ α 2. Proof. Th rquird inquality is obtaind basd on (16) and 4M 2 F +2M 2 2F. Corollary 4. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Thn, for any α 1, δ Δ α 1 with quality if and only if L(G) is rgular. χ α Δ δ α 1, In th nxt Thor w stablish a lowr bound for χ α in trs of M 1, M 2 and F. Thor 3. Lt G b a sipl connctd graph with n vrtics and dgs. Thn, for any ral α, α 1 or α 2, χ α (F +2M 2) α 1 M1 α 2. (20) If 1 α 2, th opposit inquality is valid. Equality is attaind if and only if α =1,or α =2,orL(G) is rgular. Proof. Lt p =( )anda =(a i ), i =1, 2,...,, b positiv ral nubr squncs, whr p 1 + p 2 + + p = 1. Thn, for any ral t, t 0ort 1, Jnsn s inquality holds (s [16, 18]) ( ) t a t i a i. (21) If 0 t 1 th opposit inquality is valid in (21). d( i )+2 For t = α 1, = (d( i)+2), a i = d( i )+2,i =1,...,, th inquality (21) bcos α 1 (d( i )+2) α (d( i )+2) 2. (d( i )+2) (d( i )+2)
-485- According to (18), for α =1andα = 2, this inquality transfors into χ α (F +2M 2) α 1 M 1 M1 α 1, whrfro th inquality (20) is obtaind. Equality in (21) holds if and only if t =0,ort =1,ora 1 = a 2 = = a. Thrfor quality in (20) holds if and only if α =1,orα =2,ord( 1 )+2= = d( ) + 2, i.. if L(G) is rgular. Corollary 5. Lt G b a sipl connctd graph with n vrtics and dgs. Thn, for any ral α 1, χ α 4α 1 M2 α 1 M1 α 2, with quality if α =1,orG is rgular. Rark 6. For q i = a i = d( i )+2 and b i = 1 d( i)+2, i =1, 2,...,, th inquality (10) bcos (F +2M 2 ) M1 2. Thn, for any α 1 Thrfor (20) is strongr than (7). (F +2M 2 ) α 1 M α 2 1 M α 1 α 1. In th nxt Thor w stablish a connction btwn χ α, χ α 1 and χ α 2. Thor 4. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Thn χ α (Δ + δ )χ α 1 +Δ δ χ α 2 0, (22) with quality if and only if for so k, 1 k, Δ = d( 1 )+2= = d( k )+2 d( k+1 )+2= = d( )+2=δ. (d( i )+2) α 1 Proof. For = (d( i)+2), a α 1 i = d( i )+2, i =1, 2,...,, r = δ = d( )+2 and R =Δ = d( 1 ) + 2, th inquality (8) transfors into (d( i )+2) α (d( i )+2) α 2 +Δ δ Δ + δ. (d( i )+2) α 1 (d( i )+2) α 1
-486- Fro th abov and (18) w gt χ α χ α 2 +Δ δ Δ + δ, χ α 1 χ α 1 whrfro (22) is obtaind. Corollary 6. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Thn ( ) 2 χ α χ2 α 1 Δ δ +, (23) 4χ α 2 δ Δ with quality if L(G) is rgular. Proof. According to th arithtic-gotric an inquality [16], w hav that whrfro (23) is obtaind. 2 Δ δ χ α 2 χ α χ α +Δ δ χ α 2 (Δ + δ )χ α 1, Rark 7. For α =2and α =1fro (23) w obtain F M 2 1 4 ( ) 2 Δ δ + 2M 2, (24) δ Δ and ( ) 2 M 1 2 Δ δ +, (25) 2H δ Δ whr H =2χ 1 is a haronic indx. According to Thor 1, i.. inquality (11), th following is valid: Thor 5. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Dnot by S a subst of I = {1, 2,...,} which iniizs th xprssion (d( i )+2) α 1 1 2 χ α 1. i S Thn χ α χ2 α 1 χ α 2 1+β(S) ( Δ δ δ Δ ) 2, (26) whr i S β(s) = (d( i)+2) α 1 χ α 1 Equality is attaind if L(G) is rgular. ( 1 i S (d( i)+2) α 1 ). χ α 1
-487- Proof. Th inquality (26) is obtaind fro (11) for =(d( i )+2) α 1, a i = d( i )+2, i =1, 2,...,, R =Δ = d( 1 ) + 2, and r = δ = d( )+2. Rark 8. Sinc β(s) 1, th inquality (26) is strongr than (23). Thus, for xapl, 4 for α =1,fro(26) w obtain ) 2 M 1 22 H 1+α() ( Δ δ δ whr α() = 1 ( ) 1 ( 1)+1 +1. 4 2 2 Th abov inquality is strongr than (25) whn is odd. Δ, (27) Thor 6. Lt G b a sipl connctd graph with n vrtics and 2 dgs. Thn ( ) 2 χ α χ α 2 Δ 1+α() α δ α (28) δ α Δ α with quality if L(G) is rgular. Proof. For =1,a i =(d( i )+2) α, i =1,...,, R =Δ α =(d( 1 )+2) α,andr = δ α = (d( )+2) α, according to Thor 1 w obtain (28). Rark 9. For α =1th inquality (28) rducs to (27). Acknowldgnt: This work was supportd by th Srbian Ministry for Education, Scinc and Tchnological dvlopnt. Rfrncs [1] I. Gutan, N. Trinajstić, Graph thory and olcular orbitals. Total π-lctron nrgy of altrnant hydrocarbons, Ch. Phys. Ltt. 17 (1972) 535 538. [2] K. C. Das, I. Gutan, So proprtis of th scond Zagrb indx, MATCH Coun. Math. Coput. Ch. 52 (2004) 103-112. [3] I. Gutan, On th origin of two dgr basd topological indics, Bull. Acad. Srb. Sci. Arts (Cl. Sci. Math. Natur.) 146 (2014) 39 52.
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