Multiplicative Connectivity Revan Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems

Transcription

1 Annals of Pure and Applied Mathematics Vol 6, No 2, 208, ISSN: X (P), (online) Published on 22 February 208 wwwresearchmathsciorg DOI: Annals of Multiplicative Connectivity Revan Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems VRKulli Department of Mathematics ulbarga University, ulbarga 58506, India Received 9 February 208; accepted 7 February 208 Abstract A chemical graph is a simple graph related to the structure of a chemical compound The connectivity indices are applied to measure the chemical characteristics of compounds in Chemical raph Theory In this paper, we compute the multiplicative product connectivity Revan index, multiplicative sum connectivity Revan index, multiplicative ABC Revan index for polycyclic aromatic hydrocarbons and jagged rectangle benzenoid systems Keywords: Chemical graph, multiplicative product connectivity Revan index, multiplicative sum connectivity Revan index, multiplicative ABC Revan index, polycyclic aromatic hydrocarbon, benzenoid system AMS Mathematics Subject Classification (200): 05C05, 05C07, 05C90 Introduction We consider only finite, connected, simple graph with vertex set V() and edge set E() The degree d (v) of a vertex v is the number of vertices adjacent to v Let ()( δ()) denote the maximum (minimum) degree among the vertices of The revan vertex degree r (v) of a vertex v in is defined as r (v) = () + δ() d (v) The Revan edge connecting the Revan vertices u and v will be denoted by uv For other undefined notations and terminology, the readers are referred to [] A topological index is the numeric quantity from the structural graph of a molecule Topological indices have been found to be useful in chemical documentation, isomer discrimination, structure property relationships, structure activity relationships and pharmaceutical drug design in Organic chemistry There has been considerable interest in the general problem of determining topological indices One of the best known topological index is the product connectivity Revan index, introduced by Kulli in [2] The product connectivity Revan index of a graph is defined as PR( ) = r ( u) r Recently, Revan indices were studied, for example, in [3, 4, 5, 6] 337

2 VRKulli Motivated by the definition of the product connectivity Revan index, Kulli [7] introduced the multiplicative product connectivity Revan index, multiplicative sum connectivity Revan index and multiplicative atom bond connectivity Revan index of a graph as follows: The multiplicative product connectivity Revan index of a graph is defined as PRII ( ) () r ( u) r The multiplicative sum connectivity Revan index of a graph is defined as SRII ( ) (2) r ( u) + r The multiplicative atom bond connectivity Revan index of a graph is defined as ( ) ( ) r ( u) r r u + r v 2 ABCR( ) (3) Recently several multiplicative indices were studied, for example, in [8, 9, 0,, 2, 3, 4, 5, 6 to 8, 9] Also some connectivity indices were studied, for example, in [20, 2, 22, 23, 24] In this paper, we determine the multiplicative product connectivity Revan index, the multiplicative sum connectivity Revan index and the multiplicative atom bond connectivity index for polycyclic aromatic hydrocarbons PAH n and jagged rectangle benzenoid systems B m,n For more information about aromatic hydrocarbons and benzenoid systems see [25] 2 Results for polycyclic aromatic hydrocarbons In this section, we focus on the chemical graph structure of the family of polycyclic aromatic hydrocarbons, denoted by PAH n The first three members of the family PAH n are given in Figure Figure : In the following theorem, we compute the multiplicative product connectivity Revan index of PAH n Theorem Let PAH n be the family of polycyclic aromatic hydrocarbons Then 338

3 Multiplicative Connectivity Revan Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems PRII ( PAH n ) = 3 Proof: Let = PAH n be the chemical graph in the family of polycyclic aromatic hydrocarbons We see that the vertices of are either of degree or 3 Therefore ()=3 and δ()= Thus r (u) = () + δ() d (u) = 4 d (u) By calculation, we obtain that has 6n 2 +6n vertices and 9n 2 +3n edges, see [25] In, there are two types of edges based on the degree of end vertices of each edge as follows: E 3 = {uv E() d (u) =, d (v) = 3}, E 3 = 6n E 33 = {uv E() d (u) = d (v) = 3}, E 33 = 9n 2 3n Thus there are two types of Revan edges based on the revan degree of end revan vertices of each revan edge as follows: RE 3 = {uv E() r (u) = 3, r (v) = }, RE 3 = 6n RE = {uv E() r (u) = r (v) = }, RE = 9n 2 3n To compute PRII(PAH n ), we see that ( ) PRII PAH n 3n r ( u) r r u r v r u r v RE RE n n n n = = 3 3 In the next theorem, we compute the multiplicative sum connectivity Revan index of PAH n Theorem 2 Let PAH n be the family of polycyclic aromatic hydrocarbons Then 9n( n+ ) SRII ( PAH n ) = 2 Proof: Let = PAH n be the chemical graph in the family of polycyclic aromatic hydrocarbons From equation (2) and by cardinalities of the revan edge partition of PAH n polycyclic aromatic hydrocarbons, we have SRII ( PAH n ) r ( u) + r r u + r v r u + r v RE RE 3 ( ) 2 6n 9n 3n 9n n+ = =

4 VRKulli In the following theorem, we compute the multiplicative atom bond connectivity Revan index of PAH n Theorem 3 Let PAH n be the family of polycyclic aromatic hydrocarbons Then ABCRII(PAH n ) = 0 Proof: Let = PAH n be the chemical graph in the family of polycyclic aromatic hydrocarbons, from equation (3) and by cardinalities of the revan edge partition of PAH n polycyclic aromatic hydrocarbons, we have ( ) ABCRII PAH n ( ) ( ) r ( u) r r u + r v 2 3 r u + r v 2 r u + r v 2 r u r v r u r v RE RE n n n = = Results for benzenoid systems In this section, we focus on the chemical graph structure of a jagged rectangle benzenoid system, denoted by B m,n for all m, n N Three chemical graphs of a jagged rectangle benzenoid systems are shown in Figure 2 Figure 2: In the following theorem, we compute the multiplicative product connectivity Revan index of B m, n Theorem 4 Let B m, n be the family of a jagged rectangle system Then 2n+ 4 2m+ 2n 2 6mn+ m 5n 4 PRII ( B m, n ) Proof: Let = B m, n be the chemical graph in the family of a jagged rectangle benzenoid system We see that the vertices of are either of degree 2 or 3 Therefore () = 3 and δ() = 2 Thus r (u) = () + δ() d (u) = 5 d (u) By calculation, we obtain that has 4mn+4m+2n 2 vertices and 6mn+5m+n 4 edges, see [25] In, there are three types of edges based on the degree of end vertices of each edge as follows: 340

5 Multiplicative Connectivity Revan Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems E 22 = {uv E() d (u) = d (v) = 2}, E 22 = 2n + 4 E 23 = {uv E() d (u) = 2, d (v) = 3}, E 23 = 4m + 4n 4 E 33 = {uv E() d (u) = d (v) = 3}, E 33 = 6mn + m 5n 4 Thus there are three types of Revan edges based on the revan degree of end revan vertices of each revan edge as follows: RE 33 = {uv E() r (u) = r (u) = 3}, RE 33 = 2n + 4 RE 32 = {uv E() r (u) = 3, r (u) = 2}, RE 32 = 4m + 4n 4 RE 22 = {uv E() r (u) = r (u) = 2}, RE 22 =6mn + m 5n 4 To compute PRII(B m, n ), we see that PRII ( Bm, n ) r ( u) r r u r v r u r v r u r v RE33 RE32 RE22 2n+ 4 4m+ 4n 4 6mn+ m 5n n+ 4 2m+ 2n 2 6mn+ m 5n In the following theorem, we compute the multiplicative sum connectivity Revan index of B m, n Theorem 5 Let B m,n be the family of a jagged rectangle benzenoid system Then n+ 2 2m+ 2n 2 6mn+ m 5n 4 SRII ( B m, n ) Proof: Let = B m,n be the chemical graph in the family of a jagged rectangle benzenoid system From equation (2) and by the cardinalities of the revan edge partition of B m, n, we have SRII B ( m, n ) r ( u) + r r u r v r u r v r u r v ( ) ( ) RE33 RE32 RE22 2n+ 4 4m+ 4n 4 6mn+ m 5n n+ 2 2m+ 2n 2 6mn+ m 5n

6 VRKulli In the next theorem, we compute the multiplicative atom bond connectivity Revan index of B m, n Theorem 6 Let B m,n, be the family of a jagged rectangle benzenoid system Then 2n+ 4 6mn+ 5m n 8 2 ABCRII ( B m, n ) = 3 2 Proof: Let = B m, n be a chemical graph in the family of a jagged rectangle benzenoid system From equation (3) and by cardinalities of the revan edge partition of B m, n, we have ABCRII B ( m, n ) ( ) ( ) r ( u) r r u + r v 2 2n+ 4 4m+ 4n 4 6mn+ m 5n = n + 4 6mn+ 5m n 8 2 = 3 2 REFERENCES VRKulli, College raph Theory, Vishwa International Publications, ulbarga, India (200) 2 VRKulli, On the product connectivity Revan index of certain nanotubes, Journal of Computer and Mathematical Sciences, 8(0) (207) VRKulli, Revan indices of oxide and honeycomb networks, International Journal of Mathematics and its Applications, 5(4-E) (207) VRKulli, The sum connectivity Revan index of silicate and hexagonal networks, Annals of Pure and Applied Mathematics, 4(3) (207) DOI: 5 VRKulli, Multiplicative Revan and multiplicative hyper-revan indices of certain networks, Journal of Computer and Mathematical Sciences, 8(2) (207) VRKulli, Revan indices and their polynomials of certain rhombus networks, submitted 7 VRKulli, Multiplicative connectivity Revan indices of certain families of benzenoid systems, submitted 8 VRKulli, Multiplicative connectivity indices of certain nanotubes, Annals of Pure and Applied Mathematics, 2(2) (206) VRKulli, eneral multiplicative Zagreb indices of TUC 4 C 8 [m, n] and TUC 4 [m, n] nanotubes, International Journal of Fuzzy Mathematical Archive, () (206) VRKulli, Multiplicative connectivity indices of TUC 4 C 8 [m,n] and TUC 4 [m,n] nanotubes, Journal of Computer and Mathematical Sciences, 7() (206)

7 Multiplicative Connectivity Revan Indices of Polycyclic Aromatic Hydrocarbons and Benzenoid Systems VRKulli, On multiplicative K-Banhatti and multiplicative K hyper-banhatti indices of V-Phenylenic nanotubes and nanotorus, Annals of Pure and Applied Mathematics, (2) (206) VRKulli, Multiplicative K hyper-banhatti indices and coindices of graphs, International Journal of Mathematical Archive, 7(6) (206) VRKulli, Two new multiplicative atom bond connectivity indices, Annals of Pure and Applied Mathematics, 3() (207) -7 DOI: 4 VRKulli, Multiplicative connectivity indices of nanostructures, Journal of Ultra Scientist of Physical Sciences, A 29() (207) -0 DOI: 5 VRKulli, A new multiplicative arithmetic-geometric index, International Journal of Fuzzy Mathematical Archive, 2(2) (207) DOI: 6 VRKulli, Some new multiplicative geometric-arithmetic indices, Journal of Ultra Scientist of Physical Sciencs, A, 29(2) (207) DOI: 7 VRKulli, New multiplicative inverse sum indeg index of certain benzoid systems, Journal of lobal Research in Mathematical Achieves, 4(0) (207) VRKulli, New multiplicative arithmetic-geometric indices, Journal of Ultra Scientist of Physical Sciences, A, 29(6) (207) DOI: 9 VRKulli, Edge version of multiplicative connectivity indices of some nanotubes and nanotorus, International Journal of Current Research in Science and Technology, 3() (207) VRKulli, On the product connectivity reverse index of silicate and hexagonal networks, International Journal of Mathematics and its Applications, 5(4-B) (207) VRKulli, On the sum connectivity reverse index of oxide and honeycomb networks, Journal of Computer and Mathematical Sciences, 8(9) (207) VRKulli, The product connectivity ourava index, Journal of Computer and Mathematical Sciences, 8(6) (207) VRKulli, On the sum connectivity ourava index, International Journal of Mathematical Archive, 8(7)(207) VRKulli, BChaluvaraju and HSBoregowda, Some degree based connectivity indices of Kulli cycle windmill graphs, South Asian Journal of Mathematics, 6(6) (206) VRKulli, BStone, SWang and BWei, eneralized multiplicative indices of polycyclic aromatic hydrocarbons and benzenoid systems, Z Naturforsch, 72(6) (207)