International Journal of Mathematical Archive-8(10), 2017, Available online through ISSN

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1 Interntionl Journl of Mthemticl Archive-8(10), 2017, Avilble online through wwwijminfo ISSN GENERAL ZAGREB POLYNOMIALS AND F-POLYNOMIAL OF CERTAIN NANOSTRUCTURES V R KULLI* Deprtment of Mthemtics, Gulbrg University, Gulbrg , Indi (Received On: ; Revised & Accepted On: ) ABSTRACT We introduce the generl first nd second Zgreb polynomils of moleculr grph In this pper, we compute the generl first nd second Zgreb polynomils of certin nnostructures Also we determine the first nd second Zgreb polynomils, the first nd second hyper-zgreb polynomils nd F-polynomil of these nnostructures Keywords: zgreb polynomil, hyper-zgreb polynomils, F-polynomil, nnostructure Mthemtics Subject Clssifiction: 05C05, 05C12, 05C90 1 INTRODUCTION We consider only finite, connected, undirected grphs without loops nd multiple edges Let G be grph with verte set V(G) nd edge set E(G) The degree d G (v) of verte v is the number of vertices djcent to v For other undefined nottions nd terminology, we refer to [1] In [2], Fth-Tbr defined the first nd second Zgreb polynomils of grph G They re defined s dg( u) + dg( v) M1 ( G, ), M2 ( G, ) = Recently in [3] Chluvrju et l defined the first nd second hyper-zgreb polynomils of grph G They re defined s 2 2 dg( u) + dg( v) HM1 ( G, ), HM 2 ( G, ) Motivted by these definitions, we define the generl first nd second Zgreb polynomils of grph s follows: The generl first nd second Zgreb polynomils of grph G re defined s dg( u) + dg( v) M1 ( G, ), M2 ( G, ) In [4], Furtul et l defined F-inde or forgotten topologicl inde of grph G nd it is defined s F( G) dg( u) + dg( v) The F-polynomil [5] of grph G is defined s F( G, ) dg( u) + dg( v) In this pper, the generl first nd second Zgreb polynomils, F-polynomil of some nnostructures re determined For more informtion bout nnostructures see [, 7, 8, 9] Corresponding Author: V R Kulli* Deprtment of Mthemtics, Gulbrg University, Gulbrg , Indi Interntionl Journl of Mthemticl Archive- 8(10), Oct

2 V R Kulli* / Generl Zgreb Polynomils nd F-Polynomil of Certin Nnostructures / IJMA- 8(10), Oct LINEAR [n] TETRACENE The moleculr grph of liner [n]-tetrcene is shown in Figure i n Figure-1: The moleculr grph of liner [n]-tetrcene By lgebric method, we get tht liner [n]-tetrcene hs 18n vertices nd 23n 2 edges Let T be the grph of liner [n]-tetrcene We hve three prtitions of the edge set E(T) s follows: E 4 = {uv E(T) / d T (u) = d T (v) = 2}, E 4 =, E 5 = {uv E(T) / d T (u) = 2, d T (v) = 3}, E 5 = 1n 4, E = {uv E(T) / d T (u) = d T (v) = 3}, E = 7n 4, Theorem 1: The generl first Zgreb polynomil of liner [n]-tetrcene grph T is 4 5 M1 ( T, ) = + ( 1n 4) + ( 7n 4 ) Proof: For the generl first Zgreb polynomil of liner [n]-tetrcene grph T, we hve dt ( u) + dt ( v) M1 ( T, ) uv E( T ) dt ( u) + dt ( v) dt ( u) + dt ( v) dt ( u) + dt ( v) = = + 1n 4 + 7n 4 Corollry 11: The first Zgreb polynomil of liner [n]-tetrcene grph T is M T, = M T, = + 1n 4 + 7n 4 Corollry 12: The first hyper-zgreb polynomil of liner [n]-tetrcene grph T is HM T, = M T, = + 1n 4 + 7n 4 Theorem 2: The generl second Zgreb polynomil of liner [n]-tetrcene grph T is 4 9 M2 ( T, ) = + ( 1n 4) + ( 7n 4 ) Proof: For the generl second Zgreb polynomil of liner [n]-tetrcene grph T, we hve dt ( ud ) T ( v) M2 ( T, ) uv E( T ) dt ( ud ) T ( v) dt ( ud ) T ( v) dt ( ud ) T ( v) = = + 1n 4 + 7n 4 Corollry 21: The second Zgreb polynomil of liner [n]-tetrcene grph T is M T, = M T, = + 1n 4 + 7n 4 Corollry 22: The first hyper-zgreb polynomil of liner [n]-tetrcene grph T is HM T, = M T, = + 1n 4 + 7n 4 Theorem 3: The F-polynomil of liner [n]-tetrcene grph T is F T, = + 1n 4 + 7n 4 Proof: For the F-polynomil of liner [n]-tetrcene grph T, we hve F( T, ) dt ( u) + dt ( v) 2017, IJMA All Rights Reserved 104

3 V R Kulli* / Generl Zgreb Polynomils nd F-Polynomil of Certin Nnostructures / IJMA- 8(10), Oct-2017 d ( u) + d ( v) d ( u) + d ( v) d ( u) + d ( v) T T T T T T = + + = + + 1n 4 7n 4 3 NANOSTRUCTURE F = F[p, q] The moleculr grph of nnostructure F = F[p, q] with p = 2 nd q = 4 is shown in Figure 2 Figure-2: The grph of nnostructure F = F[2, 4] Let F be the grph of nnostructure F = F[p, q] By lgebric method, we see tht F hs 18p vertices nd 27pq 2q 4p edges We hve three prtitions of the edge set E(F) s follows: E 4 = {uv E(F) / d F (u) = d F (v) = 2}, E 4 = 2q + 4, E 5 = {uv E(F) / d F (u) = 2, d F (v) = 3}, E 5 = 1p + 4q 8 E = {uv E(F) / d F (u) = d F (v) = 3}, E = 27pq 20p 8q + 4 Theorem 4: The generl first Zgreb polynomil of nnostructure grph F is 4 5 M1 ( F, ) = ( 2q+ 4) + ( 1 p+ 4q 8) + ( ) pq p q + Proof: For the generl first Zgreb polynomil of nnostructure grph F, we hve df( u) + df( v) M1 ( F, ) uv E( F ) df( u) + df( v) df( u) + df( v) df( u) + df( v) = = ( 2q+ 4) + ( 1 p+ 4q 8) + ( pq p q + ) Corollry 41: The first Zgreb polynomil of nnostructure grph F is M F, = M F, = 2q p+ 4q pq 20 p 8q+ 4 Corollry 42: The first hyper-zgreb polynomil of nnostructure grph F is HM F, = M 2 F, = 2q p+ 4q ( 27 pq 20 p 8q + 4 ) 3 Theorem 5: The generl second Zgreb polynomil of nnostructure grph F is 4 M2 ( F, ) = ( 2q+ 4) + ( 1 p+ 4q 8) + ( ) 9 pq p q + Proof: For the generl second Zgreb polynomil nnostructure grph F, we hve df( ud ) F( v) M2 ( F, ) uv E( F ) df( ud ) F( v) df( ud ) F( v) df( ud ) F( v) = = ( 2q+ 4) + ( 1 p+ 4q 8) + ( pq p q + ) , IJMA All Rights Reserved 105

4 V R Kulli* / Generl Zgreb Polynomils nd F-Polynomil of Certin Nnostructures / IJMA- 8(10), Oct-2017 Corollry 51: The second Zgreb polynomil of nnostructure grph F is M F, = M 1 F, = 2q p+ 4q 8 + ( 27 pq 20 p 8q + 4 ) 9 Corollry 52: The first hyper-zgreb polynomil of nnostructure grph F is HM F, = M 2 F, = 2q p+ 4q ( 27 pq 20 p 8q + 4 ) 81 Theorem : The F-polynomil of nnostructure grph F is 8 13 F F, = 2q p+ 4q 8 + ( 27 pq 20 p 8q + 4 ) 18 Proof: For the F-polynomil of nnostructure grph F, we hve df( u) + df( v) F( F, ) uv E( F ) d ( u) + d ( v) d ( u) + d ( v) d ( u) + d ( v) F F F F F F = + + = ( 2q+ 4) 8 + ( 1 p+ 4q 8) 13 + ( pq p q + ) NANOSTRUCTURE G = G[p, q] The moleculr grph of nnostructure G = G[p, q] with p = 2 nd q = 4 is shown in Figure 3 Figure-3: The grph of nnostructure G = G[2, 4] Let G be the grph of nnostructure G = G[p, q] By lgebric method, we see tht G hs 18pq vertices nd 27pq 4q edges We hve two prtitions of the edge set E(G) s follows: E 5 = {uv E(G) / d G (u) = 2, d G (v) = 3}, E 5 = 1p, E = {uv E(G) / d G (u) = d G (v) = 3}, E = 27pq 20p Theorem 7: The generl first Zgreb polynomil of nnostructure grph G is 5 M1 ( G, ) = 1 + ( 27 pq 20 p) Proof: For the generl first Zgreb polynomil nnostructure grph G, we hve dg( u) + dg( v) M1 ( G, ) E5 E 5 dg( u) + dg( v) dg( u) + dg( v) = + = 1 p + 27 pq 20 p Corollry 71: The first Zgreb polynomil of nnostructure grph G is 1 5 M G, = M G, = 1 p + 27 pq 20 p Corollry 72: The first hyper-zgreb polynomil of nnostructure grph G is 5 3 HM G, = M G, = 1 p + 27 pq 20 p Theorem 8: The generl second Zgreb polynomil of nnostructure grph G is 9 M 2 ( G, ) = 1 p + ( 27 pq 20 p) 2017, IJMA All Rights Reserved 10

5 V R Kulli* / Generl Zgreb Polynomils nd F-Polynomil of Certin Nnostructures / IJMA- 8(10), Oct-2017 Proof: For the generl second Zgreb polynomil of nnostructure grph G, we hve M2 ( G, ) E5 E = p 27 pq 20 p = + Corollry 81: The second Zgreb polynomil of nnostructure grph G is 1 9 M G, = M G, = 1 p + 27 pq 20 p Corollry 82: The second hyper-zgreb polynomil of nnostructure grph G is HM G, = M G, = 1 p + 27 pq 20 p Theorem 9: The F-polynomil of nnostructure grph G is F G, = 1 p + 27 pq 20 p Proof: For the F-polynomil of nnostructure grph G, we hve dg( u) + dg( v) F( G, ) d ( u) + d ( v) d ( u) + d ( v) G G G G = + E5 E = p 27 pq 20 p 5 NANOSTRUCTURE K = K[p, q] The moleculr grph of nnostructure K = K[p, q] with p = 2 nd q = 3 is shown in Figure 4 Figure-4: The grph of nnostructure K = K[2, 3] Let K be the grph of nnostructure K = K[p, q] By lgebric method, we see tht K hs 18pq vertices nd 27pq 2q edges We hve three prtitions of the edge set E(K) s follows: E 4 = {uv E(K) / d K (u) = d K (v) = 2}, E 4 = 2q E 5 = {uv E(K) / d K (u) = 2, d K (v) = 3}, E 5 = 4q E = {uv E(K) / d K (u) = d K (v) = 3}, E = 27pq 8q Theorem 10: The generl first Zgreb polynomil of nnostructure grph K is 4 5 M1 ( K, ) = 2q + 4q + ( 27 pq 8 q) Proof: For the generl first Zgreb polynomil of nnostructure grph K, we hve dk( u) + dk( v) M1 ( K, ) uv E( K ) dk( u) + dk( v) dk( u) + dk( v) dk( u) + dk( v) = = 2q + 4q + 27 pq 8 q 2017, IJMA All Rights Reserved 107

6 V R Kulli* / Generl Zgreb Polynomils nd F-Polynomil of Certin Nnostructures / IJMA- 8(10), Oct-2017 Corollry 101: The first Zgreb polynomil of nnostructure grph K is M K, = M K, = 2q + 4q + 27 pq 8 q Corollry 102: The first hyper-zgreb polynomil of nnostructure grph K is HM K, = M K, = 2q + 4q + 27 pq 8 q Theorem 11: The generl second Zgreb polynomil of nnostructure grph K is = + + ( ) 4 9 2, M K q q pq q Proof: For the generl second Zgreb polynomil of nnostructure grph K, we hve dk( ud ) K( v) M2 ( K, ) uv E( K ) dk( ud ) K( v) dk( ud ) K( v) dk( ud ) K( v) = q 4q 27 pq 8 q = + + Corollry 111: The second Zgreb polynomil of nnostructure grph K is M K, = M K, = 2q + 4q + 27 pq 8 q Corollry 112: The first hyper-zgreb polynomil of nnostructure grph K is = = + + ( ) HM K, M K, 2q 4q 27 pq 8 q Theorem 12: The F-polynomil of nnostructure grph K is F K, = 2q + 4q + 27 pq 8 q Proof: For the F- polynomil of nnostructure grph K, we hve F( K, ) dk( u) + dk( v) uv E( K ) d ( u) + d ( v) d ( u) + d ( v) d ( u) + d ( v) K K K K K K = + + = + + 2q 2q 27 pq 8 q NANOSTRUCTURE L = L[p, q] The moleculr grph of nnostructure L = L[p, q] with p = 2 nd q = 4 is shown in Figure 5 Figure-5: The grph of nnostructure L = L[2, 4] Let L be the grph of nnostructure L = L[p, q] By lgebric method, we see tht L hs 18pq vertices nd 27pq edges We hve only one prtition of the edge set E(L) s follows: E = {uv E(L) / d L (u) = d L (v) = 3}, E = 27pq 2017, IJMA All Rights Reserved 108

7 V R Kulli* / Generl Zgreb Polynomils nd F-Polynomil of Certin Nnostructures / IJMA- 8(10), Oct-2017 Theorem 13: The generl first Zgreb polynomil of nnostructure grph L, we hve M1 ( L, ) = 27 pq Proof: For the generl first Zgreb polynomil of nnostructure grph L, we hve dl u + dl v M1 ( L, ) = 27 pq uv E( L) Corollry 131: The first Zgreb polynomil of nnostructure grph L is 1 M L, = M L, = 27 pq Corollry 132: The first hyper-zgreb polynomil of nnostructure grph L is 2 3 HM L, = M L, = 27 pq Theorem 14: The generl second Zgreb polynomil of nnostructure grph L, is 9 M 2 ( L, ) = 27 pq Proof: For the generl second Zgreb polynomil of nnostructure grph L, we hve dl udl v 9 M 2 ( L, ) = 27 pq uv E( L) Corollry 141: The second Zgreb polynomil of nnostructure grph L is 1 9 M L, = M L, = 27 pq Corollry 142: The second hyper-zgreb polynomil of nnostructure grph L is 2 81 HM L, = M L, = 27 pq Theorem 15: The F-polynomil of nnostructure grph L is F L, = 27 pq 18 Proof: For the F-polynomil of nnostructure grph L, we hve L L 18 F ( L, ) d u + d v = 27 pq uv E( L) REFERENCES 1 VRKulli, College Grph Theory, Vishw Interntionl Publictions, Gulbrg, Indi (2012) 2 GH Fth-Tbr, Zgreb polynomil nd pi indices of some nnostructures, Digest Journl of Nnomterils nd Biostructures, 4(1) (2009) B Chluvrju, HSBoregowd nd SA Diwkr, Hyper-Zgreb indices nd their polynomils of some specil kinds of windmill grphs, Interntionl Journl of Advnces in Mthemtics, 2017(4), (2017) B Furtul, I Gutmn, A forgotten topologicl inde, J Mth Chem 53 (2015), N De nd SMA Nyeem, Computing the F-inde of nnostr dendrimers, Pcific Science Review A: Nturl Science nd Engineering (201) DoI: VRKulli, F-inde nd reformulted Zgreb inde of certin nnostructures, Interntionl Reserch Journl of Pure Algebr, 7(1) (2017) VRKulli, On the sum connectivity Gourv inde, Interntionl Journl of Mthemticl Archive, 8() (2017) VR Kulli, Multiplictive connectivity indices of nnostructures, Journl of Ultr Scientist of Physicl Sciences, A, 29(1), (2017) 1-10, DoI: 9 N Soleimni, MJ Nikmehr nd H A Tvllee, Computtion of the different topologicl indices of nnostructures, J Ntn Sci Foundtion SriLnk, 43(2) (2015) Source of support: Nil, Conflict of interest: None Declred [Copy right 2017 This is n Open Access rticle distributed under the terms of the Interntionl Journl of Mthemticl Archive (IJMA), which permits unrestricted use, distribution, nd reproduction in ny medium, provided the originl work is properly cited] 2017, IJMA All Rights Reserved 109