Some Inequalities for General Sum Connectivity Index

Transcription

1 MATCH Counications in Mathatical and in Coputr Chistry MATCH Coun. Math. Coput. Ch. 79 (2018) ISSN 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, Niš, Srbia {igor, (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

2 -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.

3 -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

4 -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

5 -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 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 , 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 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:

6 -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) 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

7 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 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.

8 -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 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)

9 -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

10 -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 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 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

11 -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) 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) [2] K. C. Das, I. Gutan, So proprtis of th scond Zagrb indx, MATCH Coun. Math. Coput. Ch. 52 (2004) [3] I. Gutan, On th origin of two dgr basd topological indics, Bull. Acad. Srb. Sci. Arts (Cl. Sci. Math. Natur.) 146 (2014)

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