Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations Bommanahal Basavanagoud 1, a *, Shreekant Patil 2, b

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1 Bulletin of Mathematical Sciences and Applications Submitted: ISSN: , Vol. 7, pp 0-6 Revised: doi:0.805/ Accepted: SciPress Ltd., Switzerland Online: Upper Bounds for the Modified Second Multiplicative Zagreb Index of Graph Operations Bommanahal Basavanagoud, a *, Shreekant Patil, b, Department of Mathematics, Karnatak University, Dharwad , Karnataka, India a b.basavanagoud@gmail.com, b shreekantpatil949@gmail.com Keywords: Zagreb index, hyper-zagreb index, modified second multiplicative Zagreb index, graph operations. Abstract. The modified second multiplicative Zagreb index of a connected graph G, denoted by (G), is defined as (G) = uv E(G) [d G (u) + d G (v)] [d G(u)+d G (v)] d G (z) is the degree of a vertex z in G. In this paper, we present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived.. Introduction Throughout this paper, we consider only simple connected graphs. Let G be such a graph on n vertices and m edges. We denote the vertex set and edge set of G by V(G) and E(G), respectively. Thus, V(G) = n and E(G) = m. As usual, n is said to be the order and m the size of G. If u and v are two adjacent vertices of G, then the edge connecting them will be denoted by uv. The degree of a vertex w V(G) is the number of vertices adjacent to w and is denoted by d G (w). We refer to [9] for unexplained terminology and notation. In theoretical chemistry, the physico-chemical properties of chemical compounds are often modeled by means of molecular-graph-based structure-descriptors, which are also referred to as topological indices [8, 4]. The first and second Zagreb indices, respectively, defined M (G) = u V(G) d G (u) and M (G) = uv E(G) d G (u)d G (v) are widely studied degree-based topological indices, that were introduced by Gutman and Trinajstic [7] in 97. In 03, G. H. Shirdel, H. Rezapour and A. M. Sayadi [3] introduced a new version of Zagreb index named hyper-zagreb index which is defined for a graph G as HM(G) = uv E(G) (d G (u) + d G (v)). Recently, Basavanagoud et al. [5] introduced a multiplicative version of index named modified second multiplicative Zagreb index and is defined as (G) = uv E(G) [d G (u) + d G (v)] [d G(u)+d G (v)] In this paper, we present some upper bounds for the modified second multiplicative Zagreb index of graph operations such as union, join, Cartesian product, composition and corona product of graphs are derived. Readers interested in more information on computing topological indices of graph operations can be referred to [-4, 6, 0-]. The following lemma is useful for proving our main results. SciPress applies the CC-BY 4.0 license to works we publish:

2 Bulletin of Mathematical Sciences and Applications Vol. 7 Lemma.. (Weighted AM-GM inequality) Let x, x,..., x n be nonnegative numbers and also let w, w,..., w n be nonnegative weights. Set w = w + w + +w n. If w > 0, then the inequality w x +w x + +w n x n w w w x w x w x n n holds with equality if and only if all the x k with w k > 0 are equal.. Main results All the operations considered are binary, hence we deal with two finite and simple graphs, G and G. For a given graph G i, its vertex and edge sets will be denoted by V(G i ) and E(G i ), respectively, and their cardinalities by n i and m i, respectively, i =,... Union A union G G of the graphs G and G is the graph with the vertex set V(G ) V(G ) and the edge set E(G ) E(G ). The degree d G G (u) of a vertex u is equal to the degree of u in the component G i ; i =, that contains it. The following theorem obvious by definition. Theorem.. Let G and G be two simple graphs. Then.. Join (G G ) = (G ) (G ) The join G + G of graphs G and G is a graph with the vertex set V(G ) V(G ) and edge set E(G ) E(G ) {uv: u V(G ) and v V(G )}. If u is a vertex of G + G then d G +G (u) = { d G (u) + n if u V(G ) d G (u) + n if u V(G ). Theorem.. The modified second multiplicative Zagreb index of G + G satisfies the following (G + G ) [ HM(G )+4n m +4n M (G ) ] M (G )+m n [ HM(G )+4n m +4n M (G ) M (G )+m n M (G )+m n ] M (G )+m n [ n M (G )+n M (G )+8m m +n n (n +n ) +4(n +n )(n m +m n ) ] n m +m n +n n (n +n ) () n m +m n +n n (n +n ) the equality holds in () if and only if G and G are regular graphs. Proof. By definition of modified second multiplicative Zagreb index, we have (G + G ) = uv E(G +G ) [d G +G (u) + d G +G (v)] [d G+G (u)+d G+G (v)] = uv E(G ) [d G (u) + d G (v) + n ] [d G (u)+d G (v)+n] uv E(G ) [d G (u) +d G (v) + n ] [d G (u)+d G (v)+n] u V(G ) v V(G ) [d G (u) + d G (v) + n +n ] [d G (u)+d G (v)+n +n ].

3 Volume 7 By Lemma., we have (G + G ) [ A uv E(G ) [d G (u) + d G (v) + n ] ] A [ A uv E(G ) [d G (u) + d G (v) +n ] ] A [ A u V(G ) v V(G ) [d G (u) + d G (v) + n + n ] ] A 3 () 3 A = uv E(G ) [d G (u) + d G (v) + n ] = M (G ) + m n A = uv E(G ) [d G (u) + d G (v) + n ] = M (G ) + m n A 3 = u V(G ) v V(G ) [d G (u) + d G (v) + n + n ] = n m + m n + n n (n + n ). (G + G ) [ A uv E(G ) [(d G (u) + d G (v)) + 4n + 4n (d G (u) + d G (v))]] A [ A uv E(G ) [(d G (u) + d G (v)) + 4n + 4n (d G (u) + d G (v))]] A [ A 3 u V(G ) v V(G ) [d G (u) + d G (v) + d G (u)d G (v) + (n + n ) +(n + n )(d G (u) + d G (v))]] A 3 = [ A [HM(G ) + 4n m + 4n M (G )]] A [ A [HM(G ) + 4n m + 4n M (G )]] A [ A 3 [n M (G ) + n M (G ) + 8m m + n n (n + n ) + 4(n + n )(n m +m n )]] A 3. (3) Substituting A, A and A 3 value in (3), we get the inequality (). The equality holds in () if and only if d G (u) + d G (v) + n = d G (r) + d G (s) + n for any uv, rs E(G ), d G (u) + d G (v) + n = d G (r) + d G (s) + n for any uv, rs E(G ) and d G (u) + d G (v) + n + n = d G (r) + d G (s) + n + n for any u, r V(G ), v, s V(G ) by Lemma.. This implies that the equality holds in () if and only if G and G must be regular graphs..3. Cartesian product The Cartesian product G G of graphs G and G has the vertex set V(G G ) = V(G ) V(G ) and (a, x)(b, y) is an edge of G G if and only if [a = b and xy E(G )] or [x = y and ab E(G )]. If (a, x) is a vertex of G G then d G G ((a, x)) = d G (a) + d G (x). Theorem.3. The modified second multiplicative Zagreb index of G G satisfies the following (G G ) [ 4m M (G )+n HM(G )+8m M (G ) 4m m +n M (G ) ] 4m m +n M (G ) [ 4m M (G )+n HM(G )+8m M (G ) ] 4m m +n M (G ) (4) 4m m +n M (G ) the equality holds in (4) if and only if G and G are regular graphs.

4 Bulletin of Mathematical Sciences and Applications Vol. 7 3 Proof. We have, (G G ) = (u,u )(v,v ) E(G G ) [d G G ((u, u )) u V(G ) +d G G ((v, v ))] [d G G ((u,u ))+d G G ((v,v ))] u v E(G ) [(d G (u ) + d G (u )) + (d G (u ) + d G (v ))] [(d G (u )+d G (u ))+(d G (u )+d G (v ))] d G (u )) + (d G (v ) + d G (u ))] [(d G (u )+d G (u ))+(d G (v )+d G (u ))] u V(G ) u V(G ) By Lemma., we have u V(G ) u v E(G ) [(d G (u ) + u v E(G ) [d G (u ) + (d G (u ) + d G (v ))] [d G (u )+(d G (u )+d G (v ))] u v E(G ) [d G (u ) + (d G (u ) + d G (v ))] [d G (u )+(d G (u )+d G (v))]. (G G ) [ B u V(G ) [ B u V(G ) u v E(G ) [d G (u ) + (d G (u ) + d G (v ))] ] B u v E(G ) [d G (u ) + (d G (u ) + d G (v ))] ] B (5) B = u V(G ) u v E(G ) [d G (u ) + (d G (u ) + d G (v ))] = 4m m + n M (G ) B = u V(G ) u v E(G ) [d G (u ) + (d G (u ) + d G (v ))] = 4m m + n M (G ). (G G ) [ [4m B M (G ) + n HM(G ) + 8m M (G )]] B [ B [4m M (G ) + n HM(G ) + 8m M (G )]] B. (6) Substituting B and B value in (6), we get the inequality (4). Moreover, the equality holds in (5) if and only if d G (u ) + d G (u ) + d G (v ) = d G (x ) + d G (x ) + d G (y ) for any u, x V(G ) and u v, x y E(G ) and d G (u ) + d G (u ) + d G (v ) = d G (x ) + d G (x ) + d G (y ) for any u, x V(G ) and u v, x y E(G ) by Lemma.. One can easily see that the equality in (5) if and only if d G (u ) = d G (v ) for any u, v V(G ) and d G (u ) = d G (v ) for any u, v V(G ). Hence the equality holds in (4) if and only if both G and G are regular graphs..4. Composition The composition G [G ] of graphs G and G with disjoint vertex sets V(G ) and V(G ) and edge sets E(G ) and E(G ) is the graph with vertex set V(G ) V(G ) and (u, v ) is adjacent to (u, v ) whenever [u is adjacent to u ] or [u = u and v is adjacent to v ]. If (a, b) is a vertex of G [G ] then d G [G ]((a, b)) = n d G (a) + d G (b). Theorem.4 The modified second multiplicative Zagreb index of G [G ] satisfies the following (G [G ]) [ n 4 HM(G )+m n M (G )+8m m +8n m M (G ) ] n n 3 3 M (G )+4m m n M (G )+4m m n [ 4n m M (G )+n HM(G )+8n m M (G ) ] 4n m m +n M (G ) (7) 4n m m +n M (G ) the equality holds in (7) if and only if G and G are regular graphs.

5 4 Volume 7 Proof. By definition of modified second multiplicative Zagreb index, we have (G [G ]) = (u,u )(v,v ) E(G [G ]) [d G [G ]((u, u )) +d G [G ]((v, v ))] [d G[G]((u,u ))+d G [G]((v,v ))] u v E(G ) u V(G ) v V(G ) [(n d G (u ) + d G (u )) + (n d G (v ) +d G (v ))] [(n d G (u )+d G (u ))+(n d G (v )+d G (v))] u V(G ) u v E(G ) [(n d G (u ) +d G (u )) + (n d G (u ) + d G (v ))] [(n d G (u )+d G (u ))+(n d G (u )+d G (v ))]. By Lemma., we have (G [G ]) [ C u v E(G ) u V(G ) v V(G ) [n (d G (u ) + d G (v )) + (d G (u ) + d G (v ))] ] C [ C u V(G ) u v E(G ) [n d G (u ) + (d G (u ) + d G (v ))] ] C (8) C = u v E(G ) u V(G ) v V(G ) [n (d G (u ) + d G (v )) + (d G (u ) + d G (v ))] = n 3 M (G ) + 4m m n C = u V(G ) u v E(G ) [n d G (u ) + (d G (u ) + d G (v ))] = 4n m m + n M (G ). (G [G ]) [ [n 4 C HM(G ) + m n M (G ) + 8m m + 8n m M (G )]] C [ C [4n m M (G ) + n HM(G ) + 8n m M (G )]] C (9) which gives the required result in (7) by substituting C and C value in (9). The equality holds in (8) and (9) if and only if d G (u ) = d G (v ) for any u, v V(G ) and d G (u ) = d G (v ) for any u, v V(G ) by Lemma.. Hence the equality holds in (7) if and only if both G and G are regular graphs..5. Corona product The corona product G G is defined as the graph obtained from G and G by taking one copy of G and V(G ) copies of G and then joining by an edge each vertex of the i th copy of G is named (G, i) with the i th vertex of G. If u is a vertex of G G then d G G (u) = { d G (u) + n if u V(G ) d G (u) + if u V(G, i). Theorem.5. The modified second multiplicative Zagreb index of G G satisfies the following (G G ) [ HM(G )+4n m +4n M (G ) ] M (G )+m n [ n [HM(G )+4m +4M (G )] ] n M (G )+m n M (G )+m n n M (G )+m n [ n M (G )+n M (G )+8m m +n n (n +) +4(n +)(n m +m n ) ] n m +m n +n n (n +) (0) n m +m n +n n (n +) the equality holds in (0) if and only if G and G are regular graphs.

6 Bulletin of Mathematical Sciences and Applications Vol. 7 5 Proof. We have, (G G ) = uv E(G G ) [d G G (u) + d G G (v)] [d G G (u)+d G G (v)] uv E(G ) [d G (u) + n + d G (v) + n ] [d G (u)+n +d G (v)+n ] w V(G ) u V(G ) By Lemma., we have uv E(G ) [d G (u) + + d G (v) + ] [d G (u)++d G (v)+] v V(G ) [d G (u) + n + d G (v) + ] [d G (u)+n +d G (v)+]. (G G ) [ D uv E(G ) [d G (u) + d G (v) + n ] ] D [ D w V(G ) [ D 3 u V(G ) uv E(G ) [d G (u) + d G (v) + ] ] D v V(G ) [d G (u) + d G (v) + n + ] ] D 3 () D = uv E(G ) [d G (u) + d G (v) + n ] = M (G ) + m n D = w V(G ) uv E(G ) [d G (u) + d G (v) + ] = n M (G ) + m n D 3 = u V(G ) v V(G ) [d G (u) + d G (v) + n + ] = n m + m n + n n (n + ). (G G ) [ [HM(G D ) + 4n m + 4n M (G )]] D [ [n D [HM(G ) + 4m + 4M (G )]]] D [ D 3 [n M (G ) + n M (G ) + 8m m + n n (n + ) + 4(n + )(n m + m n )]] D 3. () Substituting D, D and D 3 value in (), we get the inequality (0). The equality holds in (0) if and only if d G (u i ) = d G (u j ), u i, u j V(G ) and d G (v k ) = d G (v l ), v k, v l V(G ), that is, both G and G are regular graphs. References [] A.R. Ashrafi, T. Dos lic, A. Hamzeh, The Zagreb coindices of graph operations, Discrete Appl. Math. 58 (00) [] M. Azari, A. Iranmanesh, Some inequalities for the multiplicative sum Zagreb index of graph operations, J. Math. Inequal. 9(3) (05) [3] B. Basavanagoud, S. Patil, A note on hyper-zagreb index of graph operations, Iranian J. Math. Chem. 7() (06) [4] B. Basavanagoud, S. Patil, A note on hyper-zagreb coindex of graph operations, J. Appl. Math. Comput. (06) -9. [5] B. Basavanagoud, S. Patil, Multiplicative Zagreb indices and coindices of some derived graphs, Opuscula Math. 36(3) (06) [6] K.C. Das et al., The multiplicative Zagreb indices of graph operations, J. Inequal. Appl. (03) -4. [7] I. Gutman, N. Trinajstic, Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 7 (97)

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