ON BANHATTI AND ZAGREB INDICES

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1 JOURNAL OF THE INTERNATIONAL MATHEMATICAL VIRTUAL INSTITUTE ISSN (p) , ISSN (o) /JOURNALS / JOURNAL Vol. 7(2017), DOI: /JIMVI G Former BULLETIN OF THE SOCIETY OF MATHEMATICIANS BANJA LUKA ISSN (o), ISSN X (p) ON BANHATTI AND ZAGREB INDICES I. Gutma, V. R. Kulli, B. Chaluvaraju, ad H. S. Boregowda Abstract. Let G (V, E) be a coected graph. The Zagreb idices were itroduced as early as i They are defied as M 1 (G) d G(u) d G (v)] ad M 2 (G) d G(u)d G (v), where d G (u) deotes the degree of a vertex u. The K Bahatti idices were itroduced by Kulli i They are defied as B 1 (G) d G(u) d G (e)] ad B 2 (G) d G(u)d G (e), where meas that the vertex u ad edge e are icidet ad d G (e) deotes the degree of the edge e i G. These two types of idices are closely related. I this paper, we obtai some relatios betwee them. We also provide lower ad upper bouds for B 1 (G) ad B 2 (G) of a coected graph i terms of Zagreb idices. 1. Itroductio The graphs cosidered here are fiite, udirected, without loops ad multiple edges. Let G (V, E) be a coected graph with V (G) vertices ad E(G) m edges. The degree d G (v) of a vertex v is the umber of vertices adjacet to v. The edge coectig the vertices u ad v will be deoted by uv. Let d G (e) deote the degree of a edge e uv i G, which is defied by d G (e) d G (u) d G (v) 2. The vertices ad edges of a graph are said to be its elemets. For additioal defiitios ad otatios, the reader may refer to 11]. A molecular graph is a graph i which the vertices correspod to the atoms ad the edges to the bods of a molecule. A sigle umber that ca be computed from the molecular graph, ad used to characterize some property of the uderlyig molecule is said to be a topological idex or molecular structure descriptor. Numerous such descriptors have bee cosidered i theoretical chemistry, ad have foud some applicatios, especially i QSPR/QSAR research, see 6, 9, 17] Mathematics Subject Classificatio. 05C05; 05C07; 05C35. Key words ad phrases. Zagreb idex, hyper Zagreb idex, K Bahatti idex, K hyper Bahatti idex. 53

2 54 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA I 12], Kulli itroduced the first ad secod K Bahatti idices, itedig to take ito accout the cotributios of pairs of icidet elemets. The first K Bahatti idex B 1 (G) ad the secod K Bahatti idex B 2 (G) of a graph G are defied as B 1 (G) d G (u) d G (e)] ad B 2 (G) d G (u) d G (e) where meas that the vertex u ad edge e are icidet i G. The first ad secod K hyper Bahatti idices of a graph G are defied as HB 1 (G) d G (u) d G (e)] 2 ad HB 2 (G) d G (u) d G (e)] 2. The K hyper Bahatti idices were itroduced by Kulli i 13]. The degree based graph ivariats M 1 (G) ad M 2 (G), called Zagreb idices, were itroduced log time ago 10] ad have bee extesively studied. For the their history, applicatios, ad mathematical properties, see 2, 6, 7, 8, 15] ad the refereces cited therei. The first ad secod Zagreb idices take ito accout the cotributios of pairs of adjacet vertices. The first ad secod Zagreb idices of a graph G are defied as M 1 (G) d G (v) 2 or M 1 (G) dg (u) d G (v) ] ad v V (G) M 2 (G) d G (u) d G (v). I 14], Miličević, et al., reformulated the first Zagreb idex i terms of edgedegrees istead of vertex-degrees ad defied the respective topological idex as EM 1 (G) d G (e) 2. e E(G) Followed by the first Zagreb idex of a graph G, Furtula ad oe of the preset authors 5] itroduced the so-called forgotte topological idex F, defied as F (G) d G (u) 3 dg (u) 2 d G (v) 2]. v V (G) uv V (G) I 16], Shirdel et al., itroduced the first hyper Zagreb idex of G ad defied it as HM 1 (G) d G (u) d G (v)] Compariso of Bahatti ad Zagreb type idices Theorem 2.1. For ay graph G, the first Bahatti idex is related to the first Zagreb idex as B 1 (G) 3M 1 (G) 4m.

3 BANHATTI AND ZAGREB INDICES 55 Proof. Let G be a graph with 3 vertices ad m edges. The B 1 (G) d G (u) d G (e)] d G (u) d G (uv)] d G (u) d G (u) d G (v) 2] d G (v) d G (u) d G (v) 2] d G (v) d G (uv)] 3d G (u) 3d G (v) 4] 3M 1 (G) 4m. Theorem 2.2. For ay graph G, the secod Bahatti idex is related to the first Zagreb ad hyper Zagreb idices as B 2 (G) HM 1 (G) 2M 1 (G). Proof. Let G be a graph with 3 vertices ad m edges. The B 2 (G) d G (u) d G (e) d G (u) d G (uv) d G (u) d G (u) d G (v) 2] d G (v) d G (u) d G (v) 2] d G (v) d G (uv) d G (u) d G (v)] 2 2d G (u) d G (v)] HM 1 (G) 2M 1 (G). Theorem 2.3. Let G be a graph with 3 vertices ad m edges. EM 1 (G) HM 1 (G) 4M 1 (G) 4m. Proof. Let G be a graph with 3 vertices ad m edges. The EM 1 (G) d G (e) 2 d G (u) d G (v) 2] 2 e E(G) ( ) d G (u) d G (v)] 2 4d G (u) d G (v)] 4 HM 1 (G) 4M 1 (G) 4m. The

4 56 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA Theorem 2.4. Let G be a graph with 3 vertices ad m edges. B 1 (G) HM 1 (G) EM 1 (G) M 1 (G). Proof. EM 1 (G) HM 1 (G) 4M 1 (G) 4m HM 1 (G) M 1 (G) 3M 1 (G) 4m ] HM 1 (G) M 1 (G) B 1 (G). Theorem 2.5. Let G be a graph with 3 vertices ad m edges. B 2 (G) EM 1 (G) 2M 1 (G) 4m. Proof. EM 1 (G) HM 1 (G) 4M 1 (G) 4m HM 1 (G) 2M 1 (G) 2M 1 (G) 4m B 2 (G) 2M 1 (G) 4m. The The Corollary 2.1. Let G be a graph with 3 vertices ad m edges. The B 1 (G) B 2 (G) HM 1 (G) M 1 (G) 4m. Theorem 2.6. Let G be a graph with 3 vertices ad m edges. HB 1 (G) 2HM 1 (G) 4M 1 (G) 24m. Proof. HB 1 (G) d G (u) d G (e)] 2 d G (u) d G (uv)] 2 d G (u) d G (u) d G (v) 2] 2 d G (v) d G (u) d G (v) 2] 2 d G (v) d G (uv)] 2 2(d G (u) d G (v)) 2 4(d G (u) d G (v)) 24]. The Theorem 2.6 follows ow from the defiitios of the hyper Zagreb ad first Zagreb idices, ad the fact that E(G) has m elemets. Corollary 2.2. Let G be a graph with 3 vertices ad m edges. The B 2 (G) 1 2 HB 1(G) 12m. Proof. HB 1 (G) 2HM 1 (G) 2M 1 (G)] 24m 2B 2 (G) 24m.

5 BANHATTI AND ZAGREB INDICES 57 Corollary 2.3. Let G be a graph with 3 vertices ad m edges. The B 1 (G) 1 2 HB 1(G) EM 1 (G) M 1 (G) 12m. Proof. HB 1 (G) 2HM 1 (G) M 1 (G)] 2M 1 (G) 24m 2B 1 (G) EM 1 (G)] 2M 1 (G) 24m 2B 1 (G) 2EM 1 (G) 2M 1 (G) 24m. Theorem 2.7. Let G be a graph with 3 vertices ad m edges. HB 1 (G) 5F (G) 8M 2 (G) 12M 1 (G) 8m. Proof. HB 1 (G) d G (u) d G (e)] 2 d G (u) d G (uv)] 2 d G (u) d G (u) d G (v) 2] 2 d G (v) d G (u) d G (v) 2] 2 d G (v) d G (uv)] 2 5 d G (u) 2 d G (v) 2] 8 d G (u) d G (v) ] 12 d G (u) d G (v)] 8 5F (G) 8M 2 (G) 12M 1 (G) 8m. The I order to prove our ext result, we use the earlier established: Theorem ] Let G be a graph with 3 vertices ad m edges. The EM 1 (G) F (G) 2M 2 (G) 4M 1 (G) 4m. Corollary 2.4. Let G be a graph with 3 vertices ad m edges. The B 1 (G) F (G) 2M 2 (G) M 1 (G) EM 1 (G). Proof. From Theorem 2.8, we have EM 1 (G) F (G) 2M 2 (G) M 1 (G) (3M 1 (G) 4m) F (G) 2M 2 (G) M 1 (G) B 1 (G).

6 58 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA Corollary 2.5. Let G be a graph with 3 vertices ad m edges. The B 2 (G) F (G) 2M 2 (G) 2M 1 (G). Proof. From Theorem 2.5, we have B 2 (G) EM 1 (G) 2M 1 (G) 4m F (G) 2M 2 (G) 4M 1 (G) 4m 2M 1 (G) 4m F (G) 2M 2 (G) 2M 1 (G). 3. Bouds o Bahatti ad Zagreb type idices Theorem 3.1. For ay graph G, M 1 (G) B 1 (G). Equality is attaied if ad oly if G is totally discoected or G mk 2. Proof. Let G be a simple graph with vertices ad m edges. The by Theorem 2.1, we have B 1 (G) 3M 1 (G) 4m. Clearly M 1 (G) B 1 (G) follows. Now we prove the secod part. The graph G satisfied the give coditio B 1 (G) M 1 (G) 3M 1 (G) 4m M 1 (G) M 1 (G) 2m. Sice d G (u) 2 2m d G (u), ad (d G (u) 2 d G (u)) 0, because d G (u) 2 d G (u) 0. Thus the result follows d G (u) 2 d G (u) d G (u) 0 or d G (u) 1. Here, we use the followig existig results of the Zagreb ad K Bahatti idices of regular graph. Theorem ] Let G be a r-regular graph. The M 1 (G) r 2 ad M 2 (G) 1 2 r3. Theorem ] Let G be a r-regular graph. The B 1 (G) r(3r 2) ad B 2 (G) 2r 2 (r 1). Theorem 3.4. For ay coected graph G, B 2 (G) 4M 2 (G) 2M 1 (G). Equality is attaied if ad oly if G is a regular graph.

7 BANHATTI AND ZAGREB INDICES 59 Sice ad Proof. B 2 (G) d G (u) d G (e) d G (u) d G (u) d G (v) 2 ] d G (v) (d G (u) d G (v) 2 ] dg (u) 2 d G (v) 2 2d G (u) d G (v) ] 2M 1 (G) 4d G (u) d G (v) 2M 1 (G). d G (u) 2 d G (v) 2 2d G (u) d G (v) d G (u) 2 d G (v) 2 2d G (u) d G (v), the result follows. The equality case attais directly from Theorems 2.1, 2.2, 3.2, ad 3.3. Now, we use the followig existig results to prove our ext result. Theorem ] Let G be a simple graph with 3 vertices ad m edges. The M 1 (G) 4m2 ad M 2 (G) 4m3 2. Theorem 3.6. For ay coected graph G with 3 vertices ad m edges, B 2 (G) 8m2 (2m ) 2. Further, equality is attaied if ad oly if G is a regular graph. Proof. From Theorems , the desired result follows. Theorem 3.7. For ay coected graph G with 3 vertices ad m edges, 4m(3m ) B 1 (G) 3m 2 m. The lower boud becomes equality if ad oly if G is regular. Equality i the upper boud is attaied if ad oly if G K 1, 1 or G K 3. Proof. From Theorems 2.1 ad 3.5, bearig i mid that of M 1 (G) m(m 1), the lower ad upper bouds o B 1 (G) follow. The secod part is obvious. We ow obtai lower ad upper bouds o B 1 (G) i terms of the miimum degree δ(g) ad the maximum degree (G) of G.

8 60 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA Theorem 3.8. For ay graph G with 3 vertices ad m edges, 2m 3δ(G) 2 ] B 1 (G) 2m 3 (G) 2 ]. Further, equality i both lower ad upper bouds is attaied if ad oly if G is regular. Proof. Let G be a graph with 3 vertices ad m edges. The B 1 (G) d G (u) d G (e)] d G (u) (d G (u) d G (v) 2)] d G (v) (d G (u) d G (v) 2)] 3(d G (u) d G (v)) 4m. But 2δ(G) d G (u) d G (v) 2 (G). Bearig this i mid, 6δ(G) 3d G (u) d G (v)] 6 (G) 6δ(G) 4 3d G (u) d G (v)] 4 6 (G) 4 2m 3δ(G) 2 ] B 1 (G) 2m 3 (G) 2 ]. Further, equality i both lower ad upper bouds holds if ad oly if d G (u) d G (v) 2δ(G) 2 (G), for each uv E(G), which implies that G is a regular graph. The followig two existig results of hyper Zagreb idex to prove our ext two results i terms of δ(g) ad (G) of G. Theorem ] For ay simple graph G with 3 vertices ad m edges, HM 1 (G) δ(g) (G)]2 4mδ(G) (G) M 1(G) 2. Theorem ] For ay graph G with 3 vertices ad m edges, δ(g)m 1 (G) 2M 2 (G) HM 1 (G) (G)M 1 (G) 2M 2 (G), with equality if ad oly if G is a regular graph. Theorem For ay coected graph G with 3 vertices ad m edges, B 2 (G) Proof. From Theorem 3.9, we have δ(g) (G)]2 4mδ(G) (G) M 1(G) 2 2M 1 (G). HM 1 (G) 2M 1 (G) δ(g) (G)]2 4mδ(G) (G) M 1(G) 2 2M 1 (G)

9 BANHATTI AND ZAGREB INDICES 61 whereas from Theorem 2.2, B 2 (G) δ(g) (G)]2 4mδ(G) (G) M 1(G) 2 2M 1 (G). Theorem For ay coected graph G with 3 vertices, δ(g) 2 ] M1 (G) 2M 2 (G) B 2 (G) (G) 2 ] M1 (G) 2M 2 (G). Further, equality i both lower ad upper bouds hold if ad oly if G is regular. Proof. From Theorem 3.10, we have δ(g)m 1 (G) 2M 2 (G) 2M 1 (G) HM 1 (G) 2M 1 (G) (G)M 1 (G) 2M 2 (G) 2M 1 (G). The from Theorem 2.2, we get the desired result. Further, equality i both lower ad upper bouds will hold if ad oly if d G (u) d G (v) 2δ(G) 2 (G), for each uv E(G), which implies that G is a regular graph. Now, we use the followig existig results to prove our ext result of B 1 (T ). Theorem ] For ay tree T with 3 vertices ad m edges, 4 6 M 1 (T ) ( 1). Theorem For ay tree T with 3 vertices ad m edges, 8 14 B 1 (T ) ( 1)(3 4). Further, equality i the lower boud is attaied if ad oly if T P ad i the upper boud if ad oly if T K 1, 1. Proof. From Theorems 2.1 ad 3.13, we have B1 (T ) 4m ] ( 1) m B 1 (T ) 3( 1) 4m. Sice for ay tree T, m 1, the result follows. Further, the equality i the lower boud is attaied if ad oly if T P because B 1 (P ) Equality i the upper boud is attaied if ad oly if T K 1, 1 because B 1 (K 1, 1 ) ( 1)(3 4). I order to prove our ext result (upper boud) of B 1 (G) via M 1 (G), we apply of the Bieracki Pidek Ryll Nardzewski iequality 1].

10 62 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA Theorem ] Let a ad b be -tuples such that x a i X ad y b i Y for i 1, 2,...,. The 1 a i b i 1 a i b i 1 (X x)(y y), 4 i1 i1 with beig the greatest iteger fuctio. Equality occurs whe is eve. i1 Theorem For ay coected graph G with 3 vertices ad m edges, B 1 (G) 3 4 (G) δ(g)]2 4m (3m ). Proof. Let a i b i d G (u i ) for i 1, 2,..., with x δ(g) y ad X (G) Y. The ( 1 d G (u i ) 2 1 ) 2 2 d G (u i ) 1 (G) δ(g)]2 4 i1 i1 1 M 1(G) 1 2 (2m)2 1 (G) δ(g)]2 4 Sice we have M 1 (G) 4m2 M 1 (G) 4m2 1 B1 (G) 4m ] 4m2 3 B 1 (G) 4m 12m2 Hece the upper boud follows. 1 M 1(G) 4m (G) δ(g)]2. 1 M 1(G) 4m2 2, (G) δ(g)]2 4 (G) δ(g)] (G) δ(g)]2. I order to prove our ext result (lower boud) of B 1 (G) i terms of the miimum degree δ(g), the maximum degree (G) ad the forgotte topological idex F (G), we use of the well kow Cassel s iequality 18]. Theorem ] Let (a 1, a 2,..., a ) ad (b 1, b 2,..., b ) be positive real umbers, satisfyig the coditio 0 < l a k b k L < for each k {1, 2,..., }, where l ad L are some costats. Let (w 1, w 2,..., w ) be positive weights. The ( ) ( ) ( ) 2 w k a 2 i w k b 2 (L l)2 i w k a i b i. 4Ll i1 i1 i1

11 BANHATTI AND ZAGREB INDICES 63 Theorem For ay coected graph G with 3 vertices ad m edges, B 1 (G) 24mδ(G) (G) F (G) 4m. (δ(g) (G)) 2 Proof. Let a i d G (u i ) 3/2 ad b i d G (u i ) 1/2 with l δ(g), L (G) ad w i 1 for all 1 i. By Theorem 3.17 (Cassel s iequality), d G (u i ) 3 i1 i1 d G (u i ) F (G) 2m F (G) (δ(g) (G))2 4δ(G) (G) d G (u i ) 2 (δ(g) (G))2 M 1 (G) 4δ(G) (G) ( ) (δ(g) (G)) 2 1 B1 (G) 4m ]. 8mδ(G) (G) 3 Thus the result follows. Now, we obtai lower ad upper bouds o EM 1 (G), B 1 (G), ad B 2 (G) i terms of δ(g), (G), ad M 1 (G), usig Abel s iequality as follows. Theorem ] Let {a 1, a 2,..., a } ad {b 1, b 2,..., b } with b 1 b 2 b 0 be two seqces of real umbers ad S k a 1 a 2 a k for k 1, 2,...,. If ω mi 1 k S k ad Ω max 1 k S k, the ω b 1 a 1 b 1 a 2 b 2 a b Ω b 1. I order to prove our ext result we make use of the followig defiitio: The lie graph L(G) of the graph G is the graph whose vertices correspod to the edges of G ad two vertices i L(G) are adjacet if ad oly if the correspodig edges i G are adjacet (that is, are icidet with a commo vertex). Theorem For ay coected graph G with 3 vertices ad m edges, (3.1) (3.2) (3.3) 4(δ(G) 1) 2 EM 1 (G) 2 M 1 (G) 2m ] ( (G) 1) HM 1 (G) M 1 (G)(2 (G) 1) 4m( (G) 1) B 1 (G) HM 1 (G) M 1 (G) 4(δ(G) 1) 2 4(δ(G) 1) 2 2M 1 (G) 4m B 2 (G) 2M1 (G) 4m ] (G). Proof. Iequality (3.1): Let a i d G (e i ) with e i u i v j for i j ad b 1 b 2 b 0. Clearly, b 1 max d G (e i ) ad 2δ(G) 2 b 1 2 (G) 2, where S k a 1 a 2 a k for k 1, 2,...,.

12 64 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA Therefore ω mi 1 k S k mi 1 i d G (e i ) ω 2(δ(G) 1) ad Ω max 1 k S k max 1 i d G (e i ) S 1 d G (e i ) 2 E(L(G)) 2 2 i1 ] M 1(G) m M 1 (G) 2m. By Theorem 3.19 (Abel s iequality), we get ] d G (u i ) 2 m i1 ω b 1 a 1 b 1 a 2 b 2 a b Ω b 1 (2δ(G) 2)b 1 a 1 b 1 a 2 b 2 a b (2 (G) 2)b 1 4(δ(G) 1) 2 d G (e i ) 2 M 1 (G) 2m](2 (G) 2) i1 4(δ(G) 1) 2 EM 1 (G) 2 M 1 (G) 2m] ( (G) 1). Iequality (3.2): From (3.1) ad Theorem 2.4, we get HM 1 (G) M 1 (G)(2 (G) 1) 4m( (G) 1) B 1 (G) HM 1 (G) M 1 (G) 4(δ(G) 1) 2. Iequality (3.3): From (3.1) ad Theorem 2.5, we get 4(δ(G) 1) 2 2M 1 (G) 4m B 2 (G) (2M 1 (G) 4m) (G). Fially, we obtai the lower ad upper bouds o B 1 (G) ad B 2 (G) i terms of the umber of pedet vertices ad miimal o-pedet vertices of G. Theorem For ay (, m)-graph G with η pedet vertices ad miimal o-pedet vertex degree δ 1 (G), (3.4) (3.5) 6δ 1 (G)(m η) 3η(1 δ 1 (G)) 4m B 1 (G) 6 (G)(m η) 3η(1 (G)) 4m 4δ 1 (G)(δ 1 (G) 1)(m η) (δ 1 (G) 2 1)η B 2 (G) 4 (G)( (G) 1)(m η) ( (G) 2 1)η.

13 BANHATTI AND ZAGREB INDICES 65 Proof. Iequality (3.4): B 1 (G) d G (u) d G (e)] d G (u) (d G (u) d G (v) 2)] Thus the upper boud follows. Similarly, B 1 (G) ;d G (u),d G (v) 1 d G (v) (d G (u) d G (v) 2)] 3d G (u) d G (v)] 4 ;d G (u),d G (v) 1 ;d G (u)1 3d G (u) d G (v)] 31 d G (v)] 6 (G)(m η) 3η(1 (G)) 4m. 6δ 1 (G) ;d G (u)1 6δ 1 (G)(m η) 3η(1 δ 1 (G)) 4m. Hece the lower boud follows. Iequality (3.5): B 2 (G) d G (u) d G (e) d G (u) d G (u) d G (v) 2 ] ;d G (u),d G (v) 1 ;d G (u),d G (v) 1 ;d G (u)1 d G (u) d G (u) d G (v) 2 ] d G (v) d G (u) d G (v) 2 ] 1d G (v) 1] ;d G (u),d G (v) 1 ;d G (u)1 ;d G (u)1 4 3η(1 δ 1 (G)) d G (v)d G (v) 1] (G)(2 (G) 2) (G)(2 (G) 2) ] (G) 1 ] ;d G (u)1 (G) 1 ]. 4

14 66 GUTMAN, KULLI, CHALUVARAJU, AND BOREGOWDA Thus the upper boud follows. Similarly, B 2 (G) 2δ 1 (G) 2δ 1 (G) 2 ] ;d G (u),d G (v) 1 δ1 (G) 1 ] δ 1 (G) δ 1 (G) 1 ] ;d G (u)1 ;d G (u)1 6δ 1 (G)(m η) 3η(1 δ 1 (G)) 4m. Hece the lower boud follows. Remark 3.1. I the iequalities (3.4) ad (3.5), equality is attaied if ad oly if d G (u) d G (v) (G) δ 1 (G) for each uv E(G) with d G (u), d G (v) 1 ad d G (v) (G) δ 1 (G) for each uv E(G) with d G (u) 1. Refereces 1] M. Bieracki, H. Pidek ad C. Ryll Nardzewski. Sur ue iégalité etre des itégrales défiies, A. Uiv. Mariae Curie Sklodowska, A 4 (1950), ] B. Borovićai, K. C. Das, B. Furtula ad I. Gutma. Zagreb idices: Bouds ad extremal graphs, i: I. Gutma, B. Furtula, K. C. Das, E. Milovaović ad I. Milovaović (eds.), Bouds i Chemical Graph Theory Basics (pp ), Uiv. Kragujevac, Kragujevac, ] S. S. Dragomir, C. E. M. Pearce ad J. Sude. Abel type iequalities for complex umbers ad Gauss Polya type itegral iequalities, Math. Commu., 3(1998), ] F. Falahati Nezhad ad M. Azari. Bouds o the hyper Zagreb idex, J. Appl. Math. Iform., 34(2016), ] B. Furtula ad I. Gutma. A forgotte topological idex, J. Math. Chem., 53 (2015), ] I. Gutma. Degree based topological idices, Croat. Chem. Acta, 86(2013), ] I. Gutma ad K. C. Das. The first Zagreb idices 30 years after, MATCH Commu. Math. Comput. Chem., 50(2004), ] I. Gutma, B. Furtula, Ž. Kovijaić Vukićević ad G. Popivoda. Zagreb idices ad coidices, MATCH Commu. Math. Comput. Chem., 74(2015), ] I. Gutma ad O. E. Polasky. Mathematical Cocepts i Orgaic Chemistry, Spriger, Berli, ] I. Gutma ad N. Triajstić. Graph Theory ad molecular orbitals. Total π-electro eergy of alterat hydrocarbos, Chem. Phys. Lett., 17(1972), ] V. R. Kulli. College Graph Theory, Vishwa It. Publ., Gulbarga, ] V. R. Kulli. O K Bahatti idices of graphs, J. Comput. Math. Sci., 7(2016), ] V. R. Kulli. O K hyper Bahatti idices ad coidices of graphs, It. Res. J. Pure Algebra, 6(2016), ] A. Miličević, S. Nikolić ad N. Triajstić. O reformulated Zagreb idices, Mol. Divers., 8(2004), ] S. Nikolić, G. Kovačević, A. Miličević ad N. Triajstić. The Zagreb idices 30 years after, Croat Chem Acta, 76(2003), ] G. H. Shirdel, H. Rezapour ad A. M. Sayadi. The hyper Zagreb idex of graph operatios, Ira. J. Math. Chem., 4(2)(2013), ] R. Todeschii ad V. Cosoi. Molecular Descriptors for Chemoiformatics, Wiley VCH, Weiheim, 2009.

15 BANHATTI AND ZAGREB INDICES 67 18] G. S. Watso, G. Alpargu ad G. P. H. Stya. Some commets o six iequalities associated with the iefficiecy of ordiary least squares with oe regressor, Liear Algebra Appl., 264(1997), ] B. Zhou ad N. Triajstić. Some properties of the reformulated Zagreb idices, J. Math. Chem., 48 (2010), Received by editors ; Available olie I. Gutma: Faculty of Sciece, Uiversity of Kragujevac, Kragujevac, Serbia address: gutma@kg.ac.rs V. R. Kulli: Departmet of Mathematics, Gulbarga Uiversity, Gulbarga , Idia address: vrkulli@gmail.com B. Chaluvaraju: Departmet of Mathematics Bagalore Uiversity, Jaa Bharathi Campus, Bagalore , Idia address: bchaluvaraju@gmail.com H. S. Boregowda: Departmet of Studies ad Research i Mathematics, Tumkur Uiversity, Uiversity Costitt College Campus, Tumkur , Idia address: bgsamarasa@gmail.com