Computation of New Degree Based Topological Indices of Dutch Windmill Graph

Transcription

1 Volume 117 No , ISSN: (printed version); ISSN: (on-line version) url: ijpam.eu Computation of New Degree Based Topological Indices of Dutch Windmill Graph R. Pradeep Kumar 1 and Soner Nandappa D 1 Department of Mathematics, The National Institute of Engineering, Mysuru India. Department of Mathematics, University of Mysore, Mysuru , India. 1 pradeepr.mysore@gmail.com ndsoner@yahoo.co.in Abstract The Dutch windmill graph is denoted by D n (m) and it is the graph obtained by taking m copies of the cycle C n with a vertex in common. Topological index is a function Top from graph into real numbers with the property that T op(g) = T op(h), if G and H are isomorphic. In this paper, we compute Arithmetic - Geometric index, SK, SK 1, SK index of Dutch windmill graph. AMS Subject Classification: Primary 9.05C1, 05C6 Key Words and Phrases: Arithmetic - Geometric index, SK, SK 1, SK index. 1 Introduction The Dutch windmill graph is denoted by D n (m) and it is the graph obtained by taking m copies of the cycle C n with a vertex in common. The Dutch windmill graph is also called as friendship graph if n = 3. i.e., Friendship graph is obtained by taking m copies of the cycle C 3 with a vertex in common. Dutch windmill graph D n (m) contains (n 1)m + 1 vertices and mn edges as shown in the figure 1 to 3 [8]. 35

2 D (5) 5 D (4) 3 D (5) 4 All graphs considered in this paper are finite, connected, loop and without multiple edges. Let G = (V, E) be a graph with n vertices and m edges. The degree of a vertex u V (G) is denoted by d u and is the number of vertices that are adjacent to u. The edge connected the vertices u and v is denoted by uv, using these terminologies, certain topological indices are defined in the following manner. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariants. In an exact phrase, if graph denotes the class of all finite graphs, then a topological index is a function Top from graph into real numbers with this property that T op(g) = T op(h), if G and H are isomorphic. Obviously, the number of vertices and the number of edges are topological indices. A topological index of a chemical compound is an integer, derived following a certain rule, which can be used to characterize the chemical compound and predict certain physiochemical properties like boiling point, molecular weight, density, refractive index, and so forth [1] and []. A molecular graph is a simple graph having vertices and edges. The vertices represent non hydrogen atoms and the edges represent covalent bonds between the corresponding atoms. In particular, hydrocarbons are formed only by carbon and hydrogen atom and their molecular graphs represent the carbon skeleton of the molecule [3], [4] and [5]. Molecules and molecular compounds are often modeled by molecular graph. A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Note that hydrogen atoms are often omitted. All molecular graphs considered in this paper are finite, connected, loop less, and without multiple edges. Let G = (V, E) be a graph with vertex set V and edge set E. The degree of a vertex u E is denoted by d u and is the number of vertices that are adjacent to u. The edge connecting the vertices u and v is denoted by uv. Recently V. S. Shegehalli and R. Kanabur have introduced four new topological indices namely Arithmetic - Geometric index, SK, SK 1, SK index refer [6] to [11]. We have compute the Arithmetic - Geometric index,sk, SK 1, SK index of Dutch windmill graph. 36

3 Definitions Definition 1.1. (Arithmetic-Geometric (AG 1 ) index) Let G = (V, E) be a molecular graph and d u be the degree of the vertex u then AG 1 index of G is defined as AG 1 (G) = d u + d v where AG 1 index is considered for two distinct vertices. du d v The above equation is the sum of the ratio of the arithmetic mean and geometric mean of u and v. Definition 1.. (SK index). The SK index of a graph G = (V, E) is defined as SK(G) = d u + d v, where d u and d v are the degrees of the vertices u and v in G, respectively. Definition 1.3. (SK 1 index). The SK 1 index of a graph G = (V, E) is defined as SK 1 (G) = d u d v, where d u and d v are the degrees of the vertices u and v in G, respectively. Definition 1.4. (SK index). The SK index of a graph G = (V, E) is defined as SK (G) = ( ) du + d v, where d u and d v are the degrees of the vertices u and v in G, respectively. 3 Main Results Let G be a simple graph of order n with vertex set V = {v 1, v,..., v n } and edge set E. Theorem 1. The Arithmetic - Geometric (AG 1 ) Index of a Dutch windmill Graph D n (m) is mn m + m m + m. Proof. Consider the Dutch windmill graph D nm). ( We partition the edges of D n (m) into edges of the type E (du,dv) where uv is the edge. In D n (m) we get edges of the type E (m,). Edges of the type E (,) and E (m,) are colored in red and black respectively as shown in the figure [4]. The number of these types are given in the table 1. 37

4 figure4 D (m) n Table 1: Edge partition based on edges of end vertices of each edge. Edges of the type E (du,dv) Number of Edges E (,) (n-)m E (m,) m Consider, AG 1 (G) = d u + d v d u d v AG 1 (D n (m) + ) = e, + e m + m, m = (n )m 4 + 1) + m(m 4 4m = (n )m + m(m + 1) AG 1 (D (m) n ) = mn m + m m + m. Theorem. Proof. Consider, SK(G) = The SK Index of a Dutch windmill Graph D (m) n d u + d v SK(D n (m) ) = e, + + e m, m + = (n )m + m(m + 1) = m(n + m + 1) = m + mn m SK(D (m) n ) = m + mn m. is m +mn m. Theorem 3. The SK 1 Index of a Dutch windmill Graph D (m) n is 4m + mn 38

5 4m. Proof. Consider, SK 1 (G) = d u d v SK 1 (D n (m) ) = e, + e m, m = (n )m() + m(m) = m(n + m) = 4m + mn 4m SK 1 (D (m) n ) = 4m + mn 4m. Theorem 4. 4m 6m. Proof. Consider, SK (G) = The SK Index of a Dutch windmill Graph D (m) n is 4mn + m 3 + ( du + d v ( ) ( SK (D n (m) + m + ) = e, + e m, = (n )m() + m(m + 1) = 4mn 8m + m(m + m + 1) = 4mn + m 3 + 4m 6m SK (D (m) n ) = 4mn + m 3 + 4m 6m. ) ) References [1] 1. D. Cvetković, I. Gutman (eds.) Selected Topics on Applications of Graph spectra,(mathematical Institute Belgrade, 011). [] M. V. Diudea, I. Gutman, and J. Lorentz, Molecular Topology, Babe-Bolyai University, Cluj-Napoca, Romania, 001. [3] A. Graovac, I. Gutman, N. Trinajstić, Topological Approach to the Chemistry of Conjugated Molecules (Springer, Berlin, 1977) [4] I. Gutman, O. E. Polansky, Mathematical Concepts in Organic Chemistry (Springer, Berlin, 1986) 39

6 [5] S. M. Hosamani, S. H. Malaghan, and I. N. Cangul, The first geometricarithmetic index of graph operations, Advances and Applications in Mathematical Sciences, vol. 14, no. 6, pp , 015. [6] A. Madanshekaf and M. Moradi, The first geometric-arithmetic index of some nanostar dendrimers, Iranian Journal of Mathematical Chemistry, vol. 5, no. 1, supplement 1, pp. 1-6, 014. [7] Trinajstić, Chemical Graph Theory, Mathematical Chemistry Series, CRC Press, Boca Raton, Fla, USA, nd edition, 199. [8] Soner Nandappa D, M. R. Rajesh Kanna and R Pradeep Kumar, Narumi Katayama and Multiplicative Zagreb Indices of Dutch Windmill Graph, International Journal of Scientific and Engineering Research, Volume 7, Issue 5 (April 016), [9] V. S. Shegehalli and R. Kanabur, Arithmetic-Geometric indices of some class of Graph, Journal of Computer and Mathematical Sciences, vol. 6, no. 4, pp , 015. [10] V. S. Shegehalli and R. Kanabur, Arithmetic-Geometric indices of Path Graph, Journal of Computer and Mathematical Sciences, vol. 6, no. 1, pp. 19-4, 015. [11] V. S. Shegehalli and R. Kanabur, Computation of New Degree-Based Topological Indices of Graphene, Journal of Mathematics, vol. 5, 016, Article Id , 6 pages. * * * * * * * * * * 40

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