Journal of Mathematical Nanoscience. Sanskruti Index of Bridge Graph and Some Nanocones

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1 Joural of Mathematical Naoscieese 7 2) 2017) Joural of Mathematical Naosciece Available Olie at: Saskruti Idex of Bridge Graph ad Some Naocoes K Pattabirama * Departmet of Mathematics, Aamalai Uiversity, Aamalaiagar , Idia Academic Editor: Saeid Alikhai Abstract Saskruti idex is the importat topological idex used to test the chemical properties of chemical comopouds I this paper, first we obtai the formulae for calculatig the Saskruti idex of bridge graph ad carbo aocoes CNC k) I additio, Saskruti idex of the Lie graph of CNC k [] aocoes are obtaied Keywords Saskruti idex, bridge graph, carbo aocoes 1 Itroductio Graph theory has provided chemist with a variety of useful tools, such as topological idices Molecules ad molecular compouds are ofte modeled by molecular graph A molecular graph is a represetatio of the structural formula of a chemical compoud i terms of graph theory, whose vertices correspod to the atoms of the compoud ad edges correspod to chemical bods A graph G with vertex set VG) ad edge set EG) is coected, if there exists a coectio betwee ay pair of vertices i G For a graph G, the degree of a vertex v is the umber of edges icidet to v ad deoted by d G v) A graph ca be recogized by a umeric umber, a polyomial, a sequece of umbers or a matrix which represets the whole graph, ad these represetatios are aimed to be uiquely defied for that graph A topological idex is a umeric quatity associated with a graph which characterize the topology of graph ad is ivariat uder graph automorphism There are some major classes of topological idices such as distace based topological idices, degree based topological idices ad coutig related polyomials ad idices *Correspodig author address: pramak@gmailcom) DOI: /JMNS Shahid Rajaee Teacher Traiig Uiversity

2 K Pattabirama / Joural of Mathematical Naoscieese ) of graphs Amog these classes degree based topological idices are y ad particularly i chemistry I more precise way, a topological idex TopG) of a graph G, is a umber with the property that for every graph H isomorphic to G, TopG) = TopH) The cocept of topological idex came from work doe by Wieer [13] while he was workig o boilig poit of paraffi He amed this idex as path umber Later o, the path umber was reamed as Wieer idex The Wieer idex is the first ad most studied topological idex, both from theoretical poit of view ad applicatios, ad defied as the sum of distaces betwee all pairs of vertices i G, see for details [2, 7] Oe of the well-kow degree based topological idex is the atom-bod coectivityabc) idex of G, proposed by Estrada et al i [4], ad defied as ABCG) = dg u)d G v) 2 d G u)d G v) uv EG) Ispired by work o the ABC idex, Furtula et al [5] proposed the followig modified versio of the ABC idex ad called it as augmeted Zagreb idexazi) which is defied as AZIG) = dg u)d G 3 d G u)d G v) 2) uv EG) The predictio power is better tha the ABC idex i the study of heat of formatio for heptaes ad octaes [5] Motivated by the previous research o topological descriptors ad their applicatios, Hosamai [12] proposed a ew idex of a molecular graph G called Saskruti idex SG) which is defied as SG) = sg u)s G 3, s G u)s G v) 2) uv EG) where s G u) is the sum of degrees of all vertices adjacet to the vertex u, that is, s G u) = d G v), v N G u) where N G u) is the set of all eighbors of the give vertex u, that is, N G u) = {v VG) uv EG)} I [12], the chemical applicability of the S-idex is give ad the value of S-idex for lie graphs of subdiviso graphs of 2D-lattice, aotube ad aotorus of TUC 4 C 8 [p,q] are computed I this paper, we obtai the formulae for calculatig the Saskruti idex of bridge graph ad carbo aocoes CNC k) I additio, Saskruti idex of the lie graph of CNC k [] aocoes are obtaied 86

3 K Pattabirama / Joural of Mathematical Naoscieese ) Bridge graph I this sectio, we obtai the S-idex of bridge graph Set T = {t 1,t 2,,t k } VG) The the trucated Saskruti idex S T is defied as S {t 1,t 2,,t k } G) = S T G) = s G u)s G v) s uv EG) G u) s G v) 2 u,v/ T If T is empty, the S T G) = SG) Let G i, i {1,2,,} be certai graphs ad v i VG i ) The bridge graph expressed by G = GG 1, G 2,,G,v 1,v 2,,v ) which ca be regarded as the uio of the graphs G i, i {1,2,,} via coected edges v i v i1, i {1,2,, 1}, see Figure 1 Oe ca see that the umber of vertices ad edges of the bridge graph are respectively VG i ) ad EG i ) 1), Figure 1 The bridge graph G = GG 1,,G,v 1,,v ) The followig lemma is easily obtaied from the structure of bridge graph Lemma 21 Let G = GG 1, G 2,,G,v 1,v 2,,v ) be a bridge graph ad N G [u] = N G u) {u} for each u VG) The the followig is true: i) G = GG 1, G 2,,G,v 1,v 2,,v ) is coected if ad oly if G i, i {1,2,,} are coected ii) The degree of a vertex v VG) is d Gi v), v VG i ) ad v = v i d G v) = d Gi v) 1, v = v i ad i {1,} d Gi v) 2, v = v i ad i {2,, 1} iii) If u VG i ) ad v i / N Gi [u], the s G u) = s Gi u) where s G u) = su) ad s Gi u) = d Gi v) v N Gi u) 87

4 K Pattabirama / Joural of Mathematical Naoscieese ) Theorem 22 Set T = {t 1,t 2,,t k } VG) ad suppose v 1,v 2,,v / T For bridge graph G = GG 1, G 2,,G,v 1,v 2,,v ), we have S {t 1,t 2,,t k } G) = S T N G i [v i ] G i ) 1 sv)sv i1 ) sv i ) sv i1 ) 2 Proof By the defiitio of trucated S-idex, S {t 1,t 2,,t k } G) = = = uv EG) u,v/ T 1 su)sv) su) sv) 2 sv)sv i1 ) sv i ) sv i1 ) 2 uv EG i ) u,v/ T N Gi [v i ] uv EG i ) u,v/ T, u N Gi [v i ] S T N G i [v i ] G i ) s Gi u)s Gi v) s Gi u) s Gi v) 2 su)sv) su) sv) 2 uv EG i ) u,v/ T, u N Gi [v i ] uv EG i ) u,v/ T uv EG i ) u,v/ T, u N Gi [v i ] 1 sv)sv i1 ) sv i ) sv i1 ) 2 1 su)sv) su) sv) 2 su)sv) su) sv) 2 su)sv) su) sv) 2 sv)sv i1 ) sv i ) sv i1 ) 2 By settig = 2 i above Theorem, we obtai the followig corollary Corollary 23 For G = GG 1, G 2,v 1,v 2 )T = {t 1,t 2,,t k } VG),v 1,v 2 / T), we have S {t 1,t 2,,t k } G) = 2 S T N G i [v i ] G i ) 2 uv EG i ) u,v/ T, u N Gi [v i ] su)sv) su) sv) 2 sg1 v 1 ) d G2 v 2 ) 1)s G2 v 2 ) d G1 v 1 ) 1) s G1 v 1 ) s G2 v 2 ) d G1 v 1 ) d G2 v 2 ) Usig above corollary, we obtai the S-idex of some special molecular graphs 88

5 K Pattabirama / Joural of Mathematical Naoscieese ) Example 24 Cosider the aostar G 1 expressed i Figure 2 Oe ca check that 44) 79) 45) SG 1 ) = = ) Further, S N G 1 [v 1 ] G 1 ) = S N G 1 [v 2 ] G 1 ) = S N G 1 [v 3 ] G 1 ) = ad for ay 1 i, j 3 ad i = j we have S N G 1 [v i ] N G1 [v j ] G 1 ) = Figure 2 TThe graph of aostar dedrimer D for = 1,2,3 Now cosider the bridge graph G = GG 1, H 1,v 1,t 1 ) maifested i Figure 2 Observe that H i = G1 for i {1,2,, 1} ad G = GG 1, H 1,v 1,t 1 ), G 1 = GG 2, H 2,v 2,t 2 ), G i = GG i 1, H i1,v i1,t i1 ), G 2 = GG 1, H 1,v 1,t 1 ) 89

6 K Pattabirama / Joural of Mathematical Naoscieese ) Hece, by Corollary 23, we get the followig relatioships: SG ) = S N G 1 [v 1 ] G 1 ) S N H 1 [t 1 ] H 1 ) r, S N G 1 [v 1 ] G 1 ) = S N G 2 [v 2 ] G 2 ) S N H 2 [v 1 ] N H2 [t 2 ] H 2 ) r, S N G i [v i ] G i ) = S N G i 1 [v i1 ] G i 1 ) S N H i1 [v i ] N Hi1 [t i1 ] H i1 ) r, S N G 2 [v 2 ] G 2 ) = S N G 1 [v 1 ] G 1 ) S N H 1 [v 2 ] N H 1 [t 1 ] H 1 ) r, where 77) 55) 57) r = = ) 8 2 Combiig those relatioship stated above, we have Therefore, SG ) = S N G 1 [v 1 ] G 1 ) S N H 1 [t 1 ] H 1 ) 1 S N H i [v i 1 ] N Hi [t i ] H i ) 1)r i=2 SG ) = 2S N G 1 [v 1 ] G 1 ) 2)S G 1[v 1 ] N G1 [v 2 ] G 1 ) 1)r [ ] = ) [ ] 2) ) [ ] ) 8 2) ) [ 8 20 ] = )5 2) 2 2) 3 7 [ ] 1) 12) 8 Usig above, we have the followig theorems Theorem 25 Let G = GG 1, H 1,v 1,t 1 ) be the bridge graph preseted i Figure 2 The ) [ 8 20 ] SG ) = )5 2) 2 2) 3 7 [ ] 1) 12) 8 Theorem 26 Let D be the aostar dedrimer The ) [ 8 20 ] SD) = )5 2) 2 2) 3 7 [ ] 1) 12) 8 90

7 3 Carbo Naocoes K Pattabirama / Joural of Mathematical Naoscieese ) I this sectio, we compute the S-idex of famous carbo aocoes CNC k) Oe ca see that the umber of vertices of CNC k) is k 1) 2 ad the umber of edges of CNC k) is 2 k 1)3k 2) Before presetig our mai result i this sectio, we first see the followig two examples Example 31 Cosider the carbo aocoes CNC 3 1) show i Figure 3 This molecular structure has 15 edges, where three of them with su) = sv) = 5, six of them with su) = 7 ad sv) = 5, three of them with su) = 7 ad sv) = 9, ad three of them with su) = sv) = 9 From the defiitio of S-idex, we have 55) SCNC 3 1)) = 3 57) 79) ) ) = ) ) Figure 3 The carbo aocoes CNC 3 1) Figure 4 The carbo aocoes CNC 4 2) Example 32 Cosider the carbo aocoes CNC 4 2) show i Figure 4 This molecular structure has 48 edges, where four of them with su) = sv) = 5, eight of them with su) = 7 91

8 K Pattabirama / Joural of Mathematical Naoscieese ) ad sv) = 5, eight of them with su) = 7 ad sv) = 9, eight of them with su) = 7 ad sv) = 6, ad twety of them with su) = sv) = 9 The 55) 57) 79) SCNC 4 2)) = ) 76) 20 8 = 1072) Now we obtai the mai result for this sectio Theorem 33 Let 2 ad k 1 be positive itegers The SCNC k)) = 4k3k 1) 9 Proof From the Examples 2 ad 3, we obtai ) ) ) 5k 6) k 1) 42 55) 57) 79) SCNC k)) = 2 k ) k3k 1) 99) 2k 1) ) ) = k k3k 1) ) 4 Lie graph of CNC k [] aocoes I this sectio, we fid the S-idex of lie graph of CNC k [] aocoes The followig lemma is useful to fidig the degree of a vertex of a lie graph Let G be a graph, u VG) ad e = uv EG) The de) = du) dv) 2 Example 41 Cosider the lie graph of CNC 3 [1] I this graph we have 6 edges of su) = 6 ad sv) = 9, 3 edges of su) = sv) = 9, 6 edges of su) = 9 ad sv) = 14, 6 edges of su) = 14 ad sv) = 16 ad 3 edges of su) = sv) = 16 Thus SLCNC 3 [1])) = ) ) )3 Theorem 42 Let G be a lie graph of CNC 3 [] aocoes for > 1 The SLCNC 3 [])) = ) 375 [ 54) 3 6 [ ) ) ) ) )3 2128)3 ] ]

9 K Pattabirama / Joural of Mathematical Naoscieese ) Proof The graph G cosists of ) vertices ad 3 1)3 1) edges There are seve types of edges i EG) based o the degree sum of vertices lyig at the uit distace from ed vertices of each edge The edge partitio E 1 cotais 6 edges where su) = 6 ad sv) = 9, the edge partitio E 2 cotais 6 edges su) = 9 ad sv) = 10, the edge partitio E 3 cotais 6 edges where su) = 9 ad sv) = 14, the edge partitio E 4 cotais 6 9 edges where su) = 10 ad sv) = 10, the edge partitio E 5 cotais 6 6 edges where su) = 10 ad sv) = 14, the edge partitio E 6 cotais 6 edges where su) = 14 ad sv) = 16 ad the edge partitio E 7 cotais edges where su) = sv) = 16 su)sv) SG) = su) sv) 2 uv EG) 69) 910) 914) 1010) = ) ) 1416) 1616) 6 6) ) = ) [ ) )3 2128)3 ] [ 54) ) ) )3 ] Example 43 Cosider the lie graph of CNC 4 [1] I this graph we have 8 edges of su) = 6 ad sv) = 9, 3 edges of su) = sv) = 9, 8 edges of su) = 9 ad sv) = 14, 8 edges of su) = 14 ad sv) = 16 ad 4 edges of su) = sv) = 16 Thus SLCNC 4 [1])) = ) ) ) Theorem 44 Let G be a lie graph of CNC 4 [] aocoes for > 1 The SLCNC 4 [])) = ) 8 [ )3 8 [512 50) )3 128) ) ) )3 Proof The graph G cosists of ) vertices ad 4 1)3 1) edges There are seve types of edges i EG) based o the degree sum of vertices lyig at the uit distace from ed vertices of each edge The edge partitio E 1 cotais 8 edges where su) = 6 ad sv) = 9, the edge partitio E 2 cotais 8 edges su) = 9 ad sv) = 10, the edge partitio E 3 cotais 8 edges where su) = 9 ad sv) = 14, the edge partitio E 4 cotais 8 12 edges where su) = sv) = 10, the edge partitio E 5 cotais 8 8 edges where su) = 10 ad sv) = 14, the edge partitio E 6 cotais 8 edges where su) = 14 ad sv) = 16 ad the 93 ] ]

10 K Pattabirama / Joural of Mathematical Naoscieese ) edge partitio E 7 cotais edges where su) = sv) = 16 Thus su)sv) SG) = su) sv) 2 uv EG) 69) 910) 914) 1010) = ) ) 1416) 1616) 8 8) ) = ) 8 [512 50) )3 128)3 ] [ ) ) ) )3 ] Example 45 Cosider the lie graph of CNC k [1] I this graph we have 2k edges of su) = 6 ad sv) = 9, k edges of su) = sv) = 9, 2k edges of su) = 9 ad sv) = 14, 2k edges of su) = 14 ad sv) = 16 ad k edges of su) = sv) = 16 Thus [ SLCNC k [1])) = k ] 3375 Theorem 46 Let G be a lie graph of CNC k [] aocoes for > 1 The SLCNC k [])) = k ) 2k [512 50) )3 128)3 ] 3375 k [432 54) ) ) )3 ] Proof The graph G cosists of 2k ) vertices ad k 1)3 1) edges There are seve types of edges i EG) based o the degree sum of vertices lyig at the uit distace from ed vertices of each edge The edge partitio E 1 cotais 2k edges where su) = 6 ad sv) = 9, the edge partitio E 2 cotais 2k edges su) = 9 ad sv) = 10, the edge partitio E 3 cotais 2k edges where su) = 9 ad sv) = 14, the edge partitio E 4 cotais 2k 3k edges where su) = sv) = 10, the edge partitio E 5 cotais 2k 2k edges where su) = 10 ad sv) = 14, the edge partitio E 6 cotais 2k edges where su) = 14 ad sv) = 16 ad 94

11 K Pattabirama / Joural of Mathematical Naoscieese ) the edge partitio E 7 cotais 3k 2 2k edges where su) = sv) = 16 Thus su)sv) SG) = su) sv) 2 uv EG) 69) 910) 914) 1010) = 2k 2k 2k 2k 3k) ) 1416) 1616) 2k 2k) 2k 3k 2 2k) = k ) 2k [512 50) )3 128)3 ] 3375 k [432 54) ) ) )3 ] Refereces [1] M Baca, J Horvathova, M Mokrisova, A Suhayiova, O topological idices of fullereces, Appl Math Comput ) [2] A A Dobryi, R Etriger, I Gutma, Wieer idex of trees: theory ad applicatios, Acta Appl Math ) [3] E Estrada, Atom-bod coectivity ad the eergetic of brached alkaes, Chem Phys Lett ) [4] E Estrada, L Torres, L Rodriguez, I Gutma, A atom-bod coectivity idex: modellig the ethalpy of formatio of alkaes, Idia J Chem 37A 1998) [5] B Furtula, A Graovac, D Vukicević, Augmeted Zagreb idex, J Math Chem [6] M Ghorbai, M A Hosseizadeh, Computig ABC 4 idex of aostar dedrimers, Optoelectro Adv Mater Rapid Commu ) [7] I Gutma, O E Polasky, Mathematical Cocepts i Orgaic Chemistry, Spriger-Verlag, New York, 1986 [8] I Gutma, J Tosović, S Radeković, S Marković, O atom-bod coectivity idex ad its chemical applicability, Idia J Chem 51A 2012) [9] S Hayat, M Imra, Computatio of topological idices of certai etworks, Appl Math Comput ) [10] S A Hosseii, MB Ahmadi, I Gutma, Kragujevac trees with miimal atom-bod coectivity idex, MATCH Commu Math Comput Chem ) 5 20 [11] W Li, J Che, Q Che, T Gao, X Li, B Cai, Fast computer search for trees with miimal ABC idex based o tree degree sequeces, MATCH Commu Math Comput Chem ) [12] S M Hosamai, Computig Saskruti idex of certai aostructures, J Appl Math Comput 54 1) 2017) [13] H Wieer, Structural determiatio of paraffi boilig poits, J Am Chem Soc ) [14] B Zhou, R Xig, O atom bod coectivity idex, Z Naturforsch 66a 2011)