Narumi-Katayama and Multiplicative Zagreb Indices of Dutch Windmill Graph

International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 32 ISSN 2229-5518

Soner Nandappa D M.R. Rajesh Kanna*, R Pradeep Kumar

ABSTRACT - In this paper, we compute Narumi - Katayama index, First multiplicative Zagreb index, Modified multiplicative Zagreb index of Dutch windmill graph.

KEYWORDS - Narumi - Katayama index, First multiplicative Zagreb index, Modified multiplicative Zagreb index of Dutch windmill graph.

6, 7,8] and the references cited there in.

1. INTRODUCTION

( ) The Dutch windmill graph is denoted by and it is the We encourage the reader to consult [1, 2, 3] for basics of graph obtained by taking copies of the cycle푚 with a vertex Graph Theory, Chemical Graph Theory and molecular 푛 퐷 descriptors. in common. The Dutch windmill graph is also푛 called as friendship graph if = 3푚. i.e., Friendship graph퐶 is obtained by Definition 1.1. Let = ( , ) be a graph and degree of a vertex then Narumi - Katayama index is defined as taking m copies of the cycle with a vertex in common. Dutch 푢 ( ) 퐺 푉 퐸 푑 windmill graph 푛 contains ( 1) + 1 vertices and ( ) = ( ). It was introduced by Narumi and 3 푢 edges as shown in푚 the figure 퐶1 to 3. Katayama in [7]. 푛 푁퐾 퐺 ∏ 푑푣푣 퐺 퐷 푛 − 푚 푚푛 Definition 1.2. Let = ( , ) be a graph and degree of a vertex then First multiplicative Zagreb index is defined as 퐺 푉 퐸 푑푢 ( ) = ( )[ ( )] . The First multiplicative Zagreb 푢 index was introduced by Guttmann2 in [8]. ∏1 퐺 ∏푣∈퐸 퐺 푑푣 퐺

IJSER= ( , Definition 1.3. Let ) be a graph and degree of a

( ) ( ) ( ) vertex then Modified multiplicative Zagreb index index is 푢 defined as ( ) =퐺 푉 퐸 [ ( ) + ( )]푑. 4 5 5 ( ) 3 5 4 푢 ∗ 퐷 퐷 퐷 1 푢푣∈퐸 퐺 푢 푣 All graphs considered in this paper are finite, connected, loop ∏ 퐺 ∏ 푑 퐺 푑 퐺 and without multiple edges. Let = ( , ) be a graph with Definition 1.4. Let = ( , ) be a graph and degree of a vertices and edges. The degree of a vertex ( ) is vertex then Second multiplicative Zagreb index is defined as 퐺 푉 퐸 푑푢 denoted by and is the number퐺 of vertices푉 퐸 that are adjacent푛 ( ) = ( )[ ( ) ( )]. The second multiplicative to . The edge푚 connected the vertices and is 푢denoted ∈ 푉 퐺 by 푢 푢 Zagreb2 index푢푣 was∈퐸 퐺 introduced푢 푣 by Guttmann in [8]. , Using these푑 terminologies, certain topological indices are ∏ 퐺 ∏ 푑 퐺 푑 퐺 defined푢 in the following manner. 푢 푣 푢푣 Topological indices are numerical parameters of a 2. Main results graph which characterize its topology and are usually graph invariants. Narumi - Katayama index, First and second Theorem 1. The Narumi - Katayama index of Dutch Windmill multiplicative Zagreb index, Modified multiplicative Zagreb graph is 2 . index are the degree based molecular descriptor. For further 푛푚−푚+1 푛 ( ) results on ( ), ( ) , ( ), ( ) see the papers [4, 5, Proof. Dutch windmill graph contains ( 1) vertices ∗ 푚 1 1 2 of degree two and one vertex of푛 degree 2n. Therefore 푁퐾 ————————————————퐺 ∏ 퐺 ∏ 퐺 ∏ 퐺 퐷 푛 − 푚 Soner Nandappa D ( ) = ( ) Department of studies in Mathematics, University of Mysore, Mysuru – 푣 570006. India.Email: [email protected] 푁퐾 퐺 � 푑 ( 퐺 ) = (2푣 )2 M.R. Rajesh Kanna = 2 푛−1 푚. Post Graduate Department of Mathematics, Maharani's Science College 푛 for Women, Mysuru – 570005. India. [email protected] 푛푚−푚+1 R Pradeep Kumar 푛 Research Scholar, Department of studies in Mathematics, University of IJSER © 2016 Mysore, Mysuru – 570006. India. [email protected] http://www.ijser.org

International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 33 ISSN 2229-5518 Theorem 2. The First multiplicative index of Dutch Windmill Theorem 4. The second multiplicative index of Dutch Windmill graph is 4 . graph is 4 . 푛푚−푚+1 2 푛푚 2푚 푛 Proof . Proof . (푚 ) = [ ( ) ( )] ( ) ( ) ( )[ ( )] = (2 ) × [(2) ] (2 ) = 푢푣∈퐸 (퐺, )[푢 ( 푣) ( )] 2 2 ( 2 푛−1) 푚 ∏ ∏ 푣∈푣 퐺 푣 = 4 4 푚퐺 푑 퐺 푑푚 퐺 푚 ∏ 푑 퐺 푛 2 푛 푢푣∈퐸(2 2, )[푢 (푛 )푣 (푛 )] = 4 2 푛−1 푚 ∏ 퐷 ∏ 푑 퐷 푑 퐷 | ( , )| 푚 | 푚( , )| 푛 = (2푢푣 ×∈퐸 2)2푚 2 푢 (2 푛 × 2)푣 푛 푛푚−푚+1 2 ∏ 푑 퐷 푑 퐷 푛 = 4( ) 퐸 (42 2 ) 퐸 2푚 2 Theorem 3. The Modified first multiplicative Zagreb index of 푚 = 4 푛−2 푚 2푚 Dutch Windmill graph is 2 푛푚−2푚+2푚푚 2푚 2푚 = 4 . 푛 푚+1 . 푛푚 2푚 푚 � � 2 �� ( ) Proof: Consider the Dutch windmill graph . We partition 푚 ( ) 푚 REFERENCES the edges of into edges of the type (푛 , ) where is ( 푚) 퐷 the edge. In 푛 we get edges of the type푑 푢(푑푣, ) and ( , ). 퐷 푚 퐸 푢푣 Edges of the type ( , ) and ( , ) are colored in red and 퐷푛 퐸 2 2 퐸 2푚 2 black respectively as shown in the figure [4]. The number of [1] D.B. West. An Introduction to Graph Theory. Prentice- 퐸 2 2 퐸 2푚 2 Hall. (1996). these types are given in the table 1. [2] N. Trinajstić. Chemical Graph Theory. CRC Press, Bo ca Raton, FL. (1992). [3] R. Todeschini and V. Consonni. Handbook of Molecular Descriptors. Wiley, Weinheim. (2000). [4] Mehdi Eliasia , Ali Iranmanesha,1 , Ivan Gutmanb Multiplicative Versions of First Zagreb Index, MATCH Commun. Math. Comput. Chem. 68 (2012) 217-230. [5] J. Braun, A. Kerber, M. Meringer, C. Rucker, Similarity of molecular descriptors: The equivalence of Zagreb indices and walk counts, MATCH Figure 4 ( ) Commun. Math. Comput. Chem. 54 (2005) 163–176. [6] T. Doˇsli´c, B. Furtula, A. Graovac, I. Gutman, S. 푚 푛 Moradi, Z. Yarahmadi, On vertex– degree–based Table 1: Edge partition based퐷 on edges of end vertices of molecular structure descriptors, MATCH Commun. each edge. IJSERMath. Comput. Chem. 66 (2011) 613–626.

Edges of the type Number of Edges [7] H. Narumi, M. Katayama, Simple topological index. ( , ) A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, 푑 푢 푑 푣 ( , ) ( 2) 퐸 Mem. Fac. Engin. Hokkaido Univ. 16 (1984) 209– (2 2 , ) 2 퐸 푛 − 푚 214. 2푚 2 퐸 푚 [8] I. Gutman, Multiplicative Zagreb indices of trees, Bull. Int. Math. Virt. Inst. 1 (2011) 13–19. ( ) = ( )[ ( ) + ( )] ∗ ∏1 퐺 ∏푢푣∈퐸 퐺 푑푢 퐺 푑푣 퐺 ( ) = ( , )[ ( ) + ( )] ∗ 푚 푚 푚 ( , )[ ( ) + ( )] ∏1 퐷푛 ∏푢푣∈퐸 2 2 푑푢 퐷푛 푑푣 퐷푛 푚 푚 ∏푢푣∈퐸 2푚 2 푑푢 퐷푛 푑푣 퐷푛 = (2 + 2)| ( , )| (2 + 2)| ( , )| = 4( ) 퐸 (22 2 + 2) 퐸 2푚 2 푚 = 4(푛−2)푚 (2) ( +2푚 1) = 2 푛−2 푚 푚2푚 ( + 1) 2푚 = 22푛푚−4푚+2푚 ( +푚 1) 2푚 2푛푚−2푚 푚 2푚 = 2 푚 푚+1 2푚 푛 � � 2 ��