New Sharp Lower Bounds for the First Zagreb Index

Transcription

1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A:APPL. MATH. INFORM. AND MECH. vol. 8, 1 (016), New Shap Lowe Bouds fo the Fist Zageb Idex T. Masou, M. A. Rostami, E. Suesh, G. B. A. Xavie Abstact: The fist Zageb idex M 1 (G) is defied as the sum of squaes of the degees of the vetices. I this pape we compae ad aalyze umeous lowe bouds fo the fist Zageb idex ivolvig the umbe of vetices, the umbe of edges ad the maximum ad miimum vetex degee. I additio, we popose ew lowe boud ad coect the equality case i [E.I. Milovaović ad I.Ž. Milovaović, Shap Bouds fo the fist Zageb idex ad fist Zageb coidex, Miskolc Mathematical otes, 16 (015) ]. Keywods: Fist Zageb idex, secod Zageb idex, ivese degee. 1 Itoductio All gaphs ude discussio ae fiite, udiected ad simple. Let G =(V,E) be a simple gaph with vetices ad m edges. The degee of the vetex v i (1 i ) is deoted by d(v i ) such that d(v 1 ) d(v ) d(v ). As usual, δ ad Δ deote the miimum ad the maximum vetex degee of G. The secod maximum vetex degee is deoted by Δ. I 1987, the ivese degee was fist appeaed though cojectues of the compute pogam Gaffiti [7]. The ivese degee of a gaph G with o isolated vetices ae defied as 1 ID(G) = v V(G) d(v). Fo the ecet esults of the ivese degee, efe [, 11]. I 197, Gutma ad Tiajstić [8] exploed the study of total π-electo eegy o the molecula stuctue ad itoduced two vetex degee-based gaph ivaiats. These ivaiats ae defied as M 1 (G) = v V(G) d(v) ad M (G) = uv E(G) d(u)d(v). Oe of the most impotat ad commo mathematical popety of these ivaiats ae studyig the bouds fo the gaphs. Fo the ecet impovemets of these bouds see [4, 10] ad the efeeces ae cited theei. These bouds as usual depeds o thei stuctual vaiables (, m, Δ, δ ad simila). Mauscipt eceived Decembe 1, 015; accepted Mach 1, 016. T. Masou is with the Uivesity of Haifa, Haifa, Isael; M. A. Rostami is with the Istitute fo Compute Sciece, Fiedich Schille Uivesity Jea, Gemay; E. Suesh is with the Velammal Egieeig College, Suapet, Cheai-66, Tamil Nadu, Idia; G. B. A. Xavie is with the Saced Heat College, Tiupattu , Tamil Nadu, Idia. 11

2 1 T. Masou, M. A. Rostami, E. Suesh, G. B. A. Xavie I chemical ad mathematical liteatue umeous uppe bouds ae obtaied fo the Zageb idices, howeve oly vey few lowe bouds ae discoveed. This motivates the authos to popose some ew lowe bouds fo the fist Zageb idex ivolvig the ew paamete ivese degee ID(G) with,m,δ,δ ad δ. I additio, we compae ad aalyze ou esults with the existig lowe bouds i the liteatue so fa. Fially, we coclude that ou esults ae stoge ad ae the impovemet of the existig esults. Pelimiaies A bidegeed gaph is a gaph whose vetices have exactly two degees Δ ad δ. Let Γ be the class of gaphs such that d(v i )=δ, i =,3,...,. Γ is the special case of the Bidegeed gaphs. Let Γ ad Γ 3 be the class of gaphs, such that d(v )= = d(v 1 )=Δ, d(v )= δ with d(v 1 ) > d(v i ),i =,3,..., ad d(v i )=δ with d(v 1 ) d(v ) > d(v i ),i = 3,4,..., espectively. Next we ecall the lowe bouds fo the fist Zageb idex available i the liteatue (see [5, 9, 1, 6]). Lemma 1. Let G be a gaph with vetices ad m edges. The M 1 (G) 4m (1) equality is attaied if ad oly if G is egula. I 003, Das [3] obtaied the followig lowe boud which is bette tha Lemma 1. Lemma. Let G be a gaph with vetices ad m edges. The M 1 (G) Δ δ (m Δ δ) () with equality if ad oly if G is egula o G Γ o G Γ. I 015, Das, Xu ad Nam [4] also poposed a ew impovemet fo Lemma 1. Lemma 3. Let G be a gaph of ode ( 3), m edges with maximum degee Δ, secod maximum degee Δ ad miimum degee δ. The M 1 (G) Δ 1 with equality if ad oly if G is egula o G Γ. ( ) ( 1) (Δ δ) (3)

3 New Shap lowe bouds fo the Fist Zageb idex 13 3 Coectio of equality case Vey ecetly, E.I. Milovaović ad Ž. Milovaović [10] have poposed a ew lowe boud fo the fist Zageb idex. I additio, it was poved that Lemma 4 is bette tha Lemma 1. Lemma 4. Let G be a gaph of ode ( ) ad m edges. The M 1 (G) 4m 1 (Δ δ) (4) with equality if ad oly if G is isomophic with k-egula gaph, 1 k 1. Remak: At fist, the coclusio which elates to the equality case of (4) is wog, which we itet to complete the equality case i Lemma 4. The equality of (4) holds fo the gaphs othe tha k egula gaphs (See Gaphs G 1 ad G of Fig. 1). Let G be a gaph with vetex degees d(v 1 )=δ, d(v )= = d(v 1 )=δ 1 ad d(v )=δ. The m = M 1 (G)= fom the iequality (4), we have 4m d(v i )=(δ 1) d(v i ) =(δ ) ( )(δ 1) δ = (δ 1) 1 (Δ δ) = 1 (δ 1)(δ 1)1 (δ δ) = (δ 1) this completes that the equality of (4) holds fo the above case. Covesely, it is easy to see that, if the equality holds i (4), the G has the vetex degees d(v 1 )=δ, d(v )= = d(v 1 )=δ 1 ad d(v )=δ. Similaly, the equality of (4) holds fo the gaphs with eve ode, whose vetex degees ae d(v 1 )=k 3, d(v )= = d(v 1 )=k 1 ad d(v )=k 1 with k 1. I additio, equality holds fo d(v 1 )=k 4, d(v )= = d(v 1 )=k ad d(v )=k. I the same ituitio oe ca cojectue that the equality of (4) holds fo all gaphs with vetex degees d(v )= = d(v 1 ), it is ot tue i geeal (Refe Gaph G 3 of Fig. 1). Fially we coclude, the equality of (4) also holds if ad oly if d(v 1 )=Δ, d(v )= = d(v 1 )=Δ k ad d(v )=δ fo some 0 < k < Δ δ. Thus, it is easy to see that the boud i () is always bette tha (4) ad so we left the poof to the iteested eade.

4 14 T. Masou, M. A. Rostami, E. Suesh, G. B. A. Xavie G 1 G G 3 G 4 Fig. 1. Gaphs o 5 vetices. 4 Lowe Bouds o Fist Zageb idex Now, ou aim is to impove the existig bouds ad as well as to give some ew lowe bouds fo the fist Zageb idex i tems of,m,δ,δ ad δ. At fist we impove the classical lowe boud poposed i Lemma 1. Theoem 1. Let G be a simple gaph of ode ( 3). The M 1 (G) Δ Δ (m Δ Δ ) (5) ( ) equality holds if ad oly if G is egula o G Γ o G Γ 3. Poof. Let a 1,a,...,a ad b 1,b,...,b be ay two sequeces of eal umbes, the by Cauchy-Schwatz iequality, we get a i ( ) b i a i b i. (6) If we set =,a i = d(v i ) ad b i = 1, fo all i = 1,,,, i the above, ad usig d (v i )=m Δ Δ ad d (v i ) = M 1 (G) Δ Δ, (7) we get the equied iequality. Suppose G Γ 3, the d(v i )=δ, fo i = 3,4,...,. So ( )δ = m Δ Δ ad M1 (G)=Δ Δ ( )δ. Next, if G Γ, the d(v )= Δ = δ. So it is easy to see that if G Γ o G is egula, the equality holds. Covesely, if the equality of (5) holds, the d(v i ) = (m Δ Δ ) ( ). Usig the equality coditio of (1), we coclude that d(v i )=δ, fo i = 3,4,, ad d(v 1 ) d(v ) > δ, that is, G Γ o G Γ 3. Coollay 1. With the assumptios i Theoem 1, oe has the iequality M 1 (G) Δ ( 1) equality holds if ad oly if G is egula o G Γ. (8)

5 New Shap lowe bouds fo the Fist Zageb idex 15 Remak 1. Fo ay gaph G, the lowe boud (5) to be bette tha (1). I ode to pove this, fist we have to show that (8) is bette tha (1). Suppose, we assume that that is Δ 1 4m, ( 1)Δ (m Δ) 4m ( 1) (m Δ) 0, which leads to the cotadictio ad which fulfill ou claim. Next, by Root Mea Squae - Geometic Mea iequality, the followig iequality is always tue, that is Thus ( 1) Δ ( )(m Δ)Δ, ( 1)( )Δ ( 1)(m Δ δ) ( )(m Δ). Δ Δ (m Δ Δ ) ( ) Δ ( 1), which completes ou claim. The lowe bouds i () ad (5) ae icompaable. Namely, thee exist molecula gaph 1, 1-diethylcyclobutae fo which () is bette tha (5), ad fo 1, -diethylcyclobutae (5) is bette tha (). It is iteestig to see that fo 1, 1-dimethylcyclopopae, the lowe bouds i () ad (5) coicides togethe, othe tha equality case. Theoem. Let G be a simple gaph of ode ( 3) with o isolated vetices. The ) M1 (G) Δ Δ (m Δ Δ ) (m Δ Δ ) (ID(G) Δ 1 1 Δ ( ), (9) ad equality holds if ad oly if G is egula o G Γ o G Γ 3. Poof. Coside w 1,w,...,w be the o-egative weights, the we have the weighted vesio of the Cauchy-Schwatz iequality w i a i ( ) w i b i w i a i b i. (10)

6 16 T. Masou, M. A. Rostami, E. Suesh, G. B. A. Xavie Sice w i is o-egative, we assume that w i = x i y i with x i y i 0. So, we get x i a i x i b i ( x i a i b i ) y i a i ( ) y i b i y i a i b i 0. If we set =, a i = d(v i ) ad b i = 1, i = 1,,...,, ad sice G has o isolated 1 vetices, the we have d(v i ) 1, v i V (G). sofixx i = 1,y i = 1 i the above, we get d(v i ) ( ( ) d(v i ) 1 d (v i )) d (v i ) d (v i ) ( ) 0 (11) ( M 1 (G) Δ Δ ) ( ) (m Δ Δ ) (ID(G) 1Δ ) 1Δ (m Δ Δ ) ( ). The equality case follows the simila agumet of Theoem 1, which completes ou claim. Coollay. With the assumptios i Theoem, oe has the iequality M1 (G) Δ δ (m Δ δ) (m Δ δ)( ID(G) 1 Δ 1 ) δ ( ), (1) ad equality holds if ad oly if G is egula o G Γ o G Γ. Remak. Utilizig the iequality (11), weget (m Δ Δ ) (ID(G) 1Δ ) 1Δ ( ), this cocludes that fo ay gaph G with ( 3), ou lowe boud (9) is always bette tha the lowe boud (5). I aalogy, also we coclude that the lowe boud i (1) is stoge tha (). It is iteestig to see that, the lowe bouds i (3) ad (9) ae icompaable. Fo the gaph G 1, the lowe boud i (9) is bette tha (3) ad fo G 4, the lowe boud i (3) is bette tha (9), depicted i Fig. 1. Theoem 3. Let G be a simple gaph of ode ( 3) with o isolated vetices. The M1 (G) Δ Δ Ψ 1 (13) equality holds if ad oly if G is egula o G Γ o G Γ 3, ( ) ((m 1) Δ Δ ) (m Δ Δ ) (ID(G) ) 1 whee Ψ Δ 1 Δ 1 =.

7 New Shap lowe bouds fo the Fist Zageb idex 17 Poof. Usig (10), oe ca get ( x i a i ) 1 ( x i b i ) 1 x i a i b i ( y i a i ) 1 ( y i b i ) 1 the est of the poof follows fom the same temiology of the Theoem. y i a i b i 0, Coollay 3. With the assumptios i Theoem, oe has the iequality M1 (G) Δ δ Ψ, (14) ad equality holds if ad oly if G is egula o G Γ o G Γ, ( ((m 1) Δ δ) (m Δ δ) ( ID(G) 1 whee Ψ = Δ 1 ) ) δ Remak 3. Ou boud give by (13) is always bette tha (3). I ode to pove this, we have to show that Δ Δ Ψ 1 Δ 1 By diect obsevatio we have, Δ δ > δ, Δ > Δ 1 usig the above esults, we complete ou claim. 5 Computatioal Results ( ) ( 1) ( Δ δ Δ δ ). ad ( 1) ( ) ( ) Δ > ( 1) Δ. I this sectio, we compae five lowe bouds fo the fist Zageb idex. Fo computatioal pupose, we used GaphTea[1], a softwae tool focusig o extactig ifomatio ad visualizatio o gaphical poblems. It offes poweful ways to quey o diectly iteact with popeties of a paticula istace of a gaphical poblem. It is specially desiged fo aalyze popeties of topological idices. I Table 1, we peset the computatioal esults fo coected gaphs o = 3to = 9 vetices ad tees o = 10 to = 0 vetices. The fist thee colums cotai, the umbe of coected gaphs (tees) o vetices ad the aveage value of the fist Zageb idex M 1 (G). The ext fou goups of thee colums epeset the aveage value of the G (M 1 (G) X(G)) vetex cout ad the umbe of gaphs fo which lowe boud, the stadad deviatio the equality holds. O compaig these values alog with the Remak 3, we coclude that ou bouds (13) ad (14) has the smallest deviatio fom the fist Zageb idex ad ae stoge tha the existig esults so fa i the liteatue.

8 18 T. Masou, M. A. Rostami, E. Suesh, G. B. A. Xavie Paametes Theoem 3 Coollay 3 Lemma 3 Lemma 4 Cout Avg. Avg. Stdev. Eq. Avg. Stdev. Eq. Avg. Stdev. Eq. Avg. Stdev. Eq Table 1. Compaig the lowe bouds fo gaphs up to 9 vetices ad tees fom 10 to 0 vetices o the fist Zageb idex.

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