The multiplicative Zagreb indices of graph operations

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1 Das et al Journal of Inequalities and Applications 0, 0:90 R E S E A R C H Open Access The multiplicative Zagreb indices of graph operations Kinkar C Das, Aysun Yurttas,MugeTogan, Ahmet Sinan Cevik and Ismail Naci Cangul * * Correspondence: cangul@uludagedutr Department of Mathematics, Faculty of Arts and Science, Uludağ University, Gorukle Campus, Bursa, 6059, Turkey Full list of author information is available at the end of the article Abstract Recently, Todeschini et al Novel Molecular Structure Descriptors - Theory and Applications I, pp 7-00, 00), Todeschini and Consonni MATCH Commun Math Comput Chem 64:59-7, 00) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: G) d G v), v VG) G) d G u)d G v) uv EG) These two graph invariants are called multiplicative Zagreb indices by Gutman Bull Soc Math Banja Luka 8:7-, 0) In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs MSC: 05C05; 05C90; 05C07 Keywords: graph; multiplicative Zagreb index; graph operations Introduction Throughout this paper, we consider simple graphs which are finite, indirected graphs without loops and multiple edges Suppose G isagraphwithavertexsetvg) and an edge set EG) For a graph G, thedegreeofavertexv is the number of edges incident to v and is denoted by d G v) A topological index TopG)ofagraphG is a number with the property that for every graph H isomorphic to G, TopH)TopG) Recently, Todeschini et al, ] have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: G) d G v), G) d G u)d G v) v V G) uv EG) Mathematical properties and applications of multiplicative Zagreb indices are reported in 6] Mathematical properties and applications of multiplicative sum Zagreb indices are reportedin 7] For other undefined notations and terminology from graph theory, the readers are referred to 8] In 9, 0], Khalifeh et al computed some exact formulae for the hyper-wiener index and Zagreb indices of the join, Cartesian product, composition, disjunction and symmetric 0 Das et al; licensee Springer This is an Open Access article distributed under the terms of the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited

2 Das et al Journal of Inequalities and Applications 0, 0:90 Page of 4 difference of graphs Some more properties and applications of graph products can be seen in the classical book ] In this paper, we give some upper bounds for the multiplicative Zagreb index of various graph operations such as join, corona product, Cartesian product, composition, disjunction, etc Moreover, computations are done for some well-known graphs Multiplicative Zagreb index of graph operations We begin this section with two standard inequalities as follows Lemma AM-GM inequality) Let x, x,,x n be nonnegative numbers Then x + x + + x n n n x x x n ) holds with equality if and only if all the x k s are equal 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 x w xw xw n n ) holds with equality if and only if all the x k with w k >0are equal Let G and G be two graphs with n and n vertices and m and m edges, respectively The join G G of graphs G and G with disjoint vertex sets VG )andvg ) and edge sets EG )andeg )isthegraphuniong G together with all the edges joining VG ) and VG ) Thus, for example, K p K q K p,q,thecompletebipartitegraphwehave VG G ) n + n and EG G ) m + m + n n Theorem Let G and G be two graphs Then M G )+4m n + n n ] n M G )+4n m + n n ] n G G ) ) n and M G )+n M G )+m n ] m M G )+n M G )+m n G G ) m 4m m +n n m + m )+n n ) ] n n, 4) n n where n and n are the numbers of vertices of G and G, and m, m are the numbers of edges of G and G, respectively Moreover, the equality holds in ) if and only if both G and G are regular graphs, that is, G G is a regular graph and the equality holds in 4) if and only if both G and G are regular graphs, that is, G G is a regular graph n m ] m

3 Das et al Journal of Inequalities and Applications 0, 0:90 Page of 4 Proof Now, G G ) u i,v j ) V G G ) u i V G ) u i V G ) d G G u i, v j ) dg u i )+n ) v j V G ) dg v j )+n ) dg u i ) +n d G u i )+n ) v j V G ) and by ) this above equality is actually less than or equal to u i V G ) d G u i ) +n d G u i )+n ) v j V G ) n d G v j ) +n d G v j )+n ) n M G )+4m n + n n ] n M G )+4n m + n n ] n n ] n ] n n dg v j ) +n d G v j )+n ) Moreover, the above equality holds if and only if d G u i ) +n d G u i )+n d G u k ) +n d G u k )+n ui, u k VG ) ) and d G v j ) +n d G v j )+n d G v l ) +n d G v l )+n vj, v l VG ) ) by Lemma ), that is, for u i, u k VG )andv j, v l VG ), dg u i ) d G u k ) ) ) d G u i )+d G u k )+n and dg v j ) d G v l ) ) d G v j )+d G v l )+n ) That is, for u i, u k VG )andv j, v l VG ), we get d G u i )d G u k )andd G v j )d G v l ) Hence the equality holds in ) ifandonlyifbothg and G are regular graphs, that is, G G is a regular graph Now, since G G ) d G G u i, v j )d G G u k, v l ), we then obtain u i u k EG ) u i,v j )u k,v l ) EG G ) dg u i )+n ) dg u k )+n ) u i V G ),v j V G ) v j v l EG ) dg u i )+n ) dg v j )+n ) dg v j )+n ) dg v l )+n )

4 Das et al Journal of Inequalities and Applications 0, 0:90 Page 4 of 4 and by ) u i u k EG ) d G u i )d G u k )+n d G u i )+d G u k )) + n ) m v j v l EG ) d G v j )d G v l )+n d G v j )+d G v l )) + n ) m u i V G ),v j V G ) d G u i )d G v j )+n d G v j )+n d G u i )+n n ) ] n n 5) n n However, from the last inequality, we get ] m ] m M G )+n M G )+m n ] m M G )+n M G )+m n m u i V G ) d i v j V G ) d j + n n v i V G ) d i + n n v j V G ) d j + n n n n M G )+n M G )+m n ] m M G )+n M G )+m n ] m m 4m m +n n m + m )+n n ) ] n n n n Furthermore, for both connected graphs G and G, the equality holds in 5)iff m m ] m ] n n d G u i )d G u r )+n dg u i )+d G u r ) ) +n d G u i )d G u k )+n dg u i )+d G u k ) ) +n for any u i u r, u i u k EG ); and d G v j )d G v r )+n dg v j )+d G v r ) ) + n d G v j )d G v l )+n dg v j )+d G v l ) ) + n for any v j v r, v j v l EG )aswellas d G u i )d G v j )+n d G v j )+n d G u i )+n n d G u i )d G v l )+n d G v l )+n d G u i )+n n for any u i VG ), v j, v l VG ); and d G u i )d G v j )+n d G v j )+n d G u i )+n n d G u k )d G v j )+n d G v j )+n d G u k )+n n for any v j VG ), u i, u k VG ) by Lemma Thus one can easily see that the equality holds in 5)ifandonlyifforu i, u k VG )andv j, v l VG ), d G u i )d G u k ) and d G v j )d G v l ) Hence the equality holds in 4) ifandonlyifbothg and G are regular graphs, that is, G G is a regular graph

5 Das et al Journal of Inequalities and Applications 0, 0:90 Page 5 of 4 Example Consider two cycle graphs C p and C q Wethushave C p C q )p +) q q +) p and C p C q )p +) p+)q q +) q+)p The Cartesian product G G of graphs G and G has the vertex set VG G ) VG ) VG )andu i, v j )u k, v l ) is an edge of G G if either u i u k and v j v l EG ), or u i u k EG )andv j v l Theorem Let G and G be two connected graphs Then i) ] n M G )+n M G )+8m m n n G G ) 6) n n The equality holds in 6) if and only if G G is a regular graph ii) ) n m G G ) n M n m ) n G )+4m m m n m ) n m n M G )+4m m ) n m 7) Moreover, the equality holds in 7) if and only if G G is a regular graph Proof By the definition of the first multiplicative Zagreb index, we have G G ) u i,v j ) V G G ) On the other hand, by ) u i V G ) v j V G ) dg u i )+d G v j ) ) dg u i )+d G v j ) ) u i V G ) v j V G ) d G u i ) + d G v j ) +d G u i )d G v j )) ] n n 8) n n But as u i V G ) d G u i ) M G )and v j V G ) d G v j ) M G ), the last statement in 8)islessthanorequalto u i V G ) d G u i ) v j V G ) + v j V G ) d G v j ) +d G u i ) v j V G ) d G v j )) ] n n n n which equals to ] n M G )+n M G )+8m m n n n n

6 Das et al Journal of Inequalities and Applications 0, 0:90 Page 6 of 4 Moreover, the equality holds in 8) ifandonlyifd G u i )+d G v j )d G u k )+d G v l ) for any u i, v j ), u k, v l ) VG G ) by Lemma SincebothG and G are connected graphs, one can easily see that the equality holds in 8) ifandonlyifd G u i )d G u k ), u i, u k VG )andd G v j )d G v l ), v j, v l VG ) Hence the equality holds in 6) ifand only if both G and G are regular graphs, that is, G G is a regular graph This completes the first part of the proof By the definition of the second multiplicative Zagreb index, we have G G ) u i,v j )u k,v l ) EG G ) This actually can be written as G G ) or, equivalently, u i V G ) v j v l EG ) G G ) After that, by )weget v j V G ) u i u k EG ) u i V G ) v j V G ) G G ) Moreover, since v j V G ) u i V G ) u i V G ) v j V G ) dg u i )+d G v j ) ) d G u k )+d G v l ) ) dg u i )+d G v j ) ) d G u i )+d G v l ) ) dg u i )+d G v j ) ) d G u k )+d G v j ) ) dg u i )+d G v j ) ) d G v j ) dg u i )+d G v j ) ) d G u i ) v j V G ) d G v j )d G u i )+d G v j )) m ] m u i V G ) d G u i )d G u i )+d G v j )) m ] m 9) d G u i )m, u i V G ) d G u i ) M G ), u i V G ) d G v j )m and v j V G ) d G v j ) M G ) v j V G ) By ) the final statement in 9)becomes u i V G ) ] M G )+m d G u i ) m m v j V G ) u i V G ) M G )+m d G u i )) m ) n m n M G )+m d G v j ) ] n m m ] m

7 Das et al Journal of Inequalities and Applications 0, 0:90 Page 7 of 4 v j V G ) M G )+m d G v j )) ] n m 0) m ) n m n n m ) n m n M G )+4m m ) n m n m ) n m n M G )+4m m ) n m Hence the second part of the proof is over The equality holds in 9) and0) ifandonlyifd G v j )d G v l ) for any v j, v l VG ) and d G u i )d G u k ) for any u i, u k VG ) by Lemmas and Hence the equality holds in 7)ifandonlyifbothG and G are regular graphs, that is, G G is a regular graph This completes the proof Example Consider a cycle graph C p and a complete graph K q Wethushave C p K q )q +) pq and C p K q )q +) q+)pq The corona product G G of two graphs G and G is defined to be the graph Ɣ obtained by taking one copy of G which has n vertices) and n copies of G,andthen joining the ith vertex of G to every vertex in the ith copy of G, i,,,n Let G V, E) andg V, E) be two graphs such that VG ){u, u,,u n }, EG ) m and VG ){v, v,,v n }, EG ) m Then it follows from the definition of the corona product that G G has n + n ) vertices and m + n m + n n edges, where VG G ){u i, v j ), i,,,n ; j 0,,,,n } and EG G ) {u i, v 0 ), u k, v 0 )), u i, u k ) EG )} {u i, v j ), u i, v l )), v j, v l ) EG ), i,,,n } {u i, v 0 ), u i, v l )), l,,,n, i,,,n }ItisclearthatifG is connected, then G G is connected, and in general G G is not isomorphic to G G Theorem The first and second multiplicative Zagreb indices of the corona product are computed as follows: i) ii) G G ) n n nn n M G ) n M G )+4m + n ) n n, ) M G )+n M G )+n ] m M G )+M G )+ G G ) m m ] n m 4m m + n n +m ] n +m n n n n, ) n n where M G i ) and M G i ) are the first and second Zagreb indices of G i, where i,,respectively Moreover, both equalities in ) and ) hold if and only if G G is a regular graph

8 Das et al Journal of Inequalities and Applications 0, 0:90 Page 8 of 4 Proof By the definition of the first multiplicative Zagreb index, we have G G ) u i,v j ) V G G ) u i V G ) u i V G ) d G G u i, v j ) dg u i )+n ) u i V G ) v j V G ) dg u i ) +n d G u i )+n ) v j V G ) dg v j ) +d G v j )+ ) ] n dg v j )+ ) u i V G ) d G u i ) +n d G u i )+n ) ] n n v j V G ) d G v j ) +d G v j )+) ] n n by ) ) n n nn n n M G )+4n m + n n ) n ) n n M G )+4m + n The equality holds in ) ifandonlyifd G u i )d G u k ), u i, u k VG )andd G v j ) d G v l ), v j, v l VG ), that is, both G and G are regular graphs, that is, G G is a regular graph By the definition of the second multiplicative Zagreb index, we have G G ) d G G u i, v j )d G G u k, v l ) u i,v j )u k,v l ) EG G ) ) ) dg u i )+n dg u k )+n u i u k EG ) u i V G ) v j V G ) ) dg u i )+n dg v j )+ ) dg v j )+ ) d G v l )+ ) u i V G ) v j v l EG ) dg u i )d G u k )+n dg u i )+d G u k ) ) + n ) u i u k EG ) ) ] n dg u i )+n dg v j )+ )] n u i V G ) v j V G ) dg v j )d G v l )+ d G v j )+d G v l ) ) + )] n v j v l EG ) M G )+n M G )+n m ] m m + n n m n ] n n ] m + n n n ] M G )+M G )+m n m by ) n m

9 Das et al Journal of Inequalities and Applications 0, 0:90 Page 9 of 4 The above equality holds if and only if d G u i )d G u k ) for any u i, u k VG )andd G v j ) d G v l ) for any v j, v l VG ), that is, both G and G are regular graphs, which implies that G G is a regular graph This completes the proof Example C p K q )q pq q +) p and C p K q )q pq q +) pq+) The composition also called lexicographic product ]) G G G ]ofgraphsg and G with disjoint vertex sets VG )andvg ) and edge sets EG )andeg )isthegraph with a vertex set VG ) VG )andu i, v j )isadjacenttou k, v l )whenever either u i is adjacent to u k, or u i u k and v j is adjacent to v l Theorem 4 The first and second multiplicative Zagreb indices of the composition G G ] of graphs G and G are bounded above as follows: i) ii) G G ] ) n n n ) n n M G )+8n m m + n M G ) ] n n, 4) G G ] ) m n n m ) n m M G )+n m M G )+n M G ) ] n m n n m ) m n M G )+m M G )+m n M G ) ] n m, 5) where M G i ) and M G i ) are the first and second Zagreb indices of G i, where i,moreover, the equalities in 4) and 5) hold if and only if G G is a regular graph Proof By the definition of the first multiplicative Zagreb index, we have G G ] ) u i,v j ) V G G ]) u i V G ) v j V G ) d G G ]u i, v j ) dg u i )n + d G v j ) ) u i V G ) v j V G ) n d G u i ) +n d G u i )d G v j )+d G v j ) ) ] n n 6) n n n n ) n n n M G )+8n m m + n M G ) ] n n The equality holds in 6) ifandonlyifd G u i )d G u k ), u i, u k VG )andd G v j ) d G v l ), v j, v l VG ) by Lemma ), that is, both G and G are regular graphs, that is, G G is a regular graph

10 Das et al Journal of Inequalities and Applications 0, 0:90 Page 0 of 4 By the definition of the second multiplicative Zagreb index, we have G G ] ) u i,v j )u k,v l ) EG G ]) u i V G ) v j v l EG ) u i u k EG ) v j V G ) u i V G ) d G G u i, v j )d G G ]u k, v l ) dg u i )n + d G v j ) ) d G u i )n + d G v l ) ) dg u i )n + d G v j ) ) d G u k )n + d G v j ) )] n m n d G u i ) + n d G u i )M G )+M G ) u i u k EG ) m n ] d G u i )d G u k )+M G )+m n d G u i )+d G u k )) n 7) m n m n M ] G )+n m M G )+n M G ) n m m n n ] M G )+m M G )+m n M G ) n m, 8) n ) m n m which gives the required result in 5) The equality holds in 7) and8) ifandonlyifd G u i )d G u k ), u i, u k VG )and d G v j )d G v l ), v j, v l VG ) by Lemma ), that is, both G and G are regular graphs, that is, G G is a regular graph n ] m Example 4 C pc q ]) pq q +) pq and C pc q ]) pqq+) q +) pqq+) The disjunction G G of graphs G and G is the graph with a vertex set VG )VG ) and u i, v j )isadjacenttou k, v l )wheneveru i u k EG )orv j v l EG ) Theorem 5 The first and second multiplicative Zagreb indices of the disjunction are computed as follows: i) G G ) n n n ) n n M G )+n M G )+M G )M G ) +8n n m m 4n m M G ) 4n m M G ) ] n n, 9) ii) G G ) M G )n + M G ) 4n m )+M G )n 4n ] m )+8n n m m Q, 0) Q

11 Das et al Journal of Inequalities and Applications 0, 0:90 Page of 4 where Q u i V G ) v j V G ) P n m + n m m m ) and M G i ) is the first Zagreb index of G i, i,moreover, the equalities in 9) and 0) hold if and only if G G is a regular graph Proof We have d G G u i, v j )n d G u i )+n d G v j ) d G u i )d G v j ) By the definition of the first multiplicative Zagreb index, we have G G ) u i,v j ) V G G ) u i V G ) v j V G ) d G G u i, v j ) n d G u i )+n d G v j ) d G u i )d G v j ) ) u i V G ) v j V G ) n d G u i )+n d G v j ) d G u i )d G v j )) ] n n ) n n n n n ) n n M G )+n M G )+M G )M G ) +8n n m m 4n m M G ) 4n m M G ) ] n n The equality holds in ) ifandonlyifd G u i )d G u k ), u i, u k VG )andd G v j ) d G v l ), v j, v l VG ) by Lemma ), that is, both G and G are regular graphs, that is, G G is a regular graph By the definition of the second multiplicative Zagreb index, we have G G ) d G G u i, v j )d G G u k, v l ) where u i,v j )u k,v l ) EG G ) u i V G ) v j V G ) P P, P n d G u i )+n d G v j ) d G u i )d G v j ) Using the weighted arithmetic-geometric mean inequality in ), G G )isless than or equal to where u i V G ) v j V G ) n d G u i )+n d G v j ) d G u i )d G v j )) ] u i V G ) v j V G ) P u i V G ) v j V G ) P ) M G )n + M G ) 4n m )+M G )n 4n ] m )+8n n m m Q, Q Q u i V G ) v j V G ) P n m + n m m m ) Hence the first part of the proof is over

12 Das et al Journal of Inequalities and Applications 0, 0:90 Page of 4 The equality holds in ) ifandonlyifd G u i )d G u k ), where u i, u k VG )and d G v j )d G v l ), where v j, v l VG ) by Lemma ), that is, both G and G are regular graphs, and so the graph G G is regular Example 5 K p C q )pq q +) pq and K p C q )pq q +) pqpq q+) The symmetric difference G G of two graphs G and G is the graph with a vertex set VG ) VG )inwhichu i, v j )isadjacenttou k, v l )wheneveru i is adjacent to u k in G or v i is adjacent to v l in G,butnotbothThedegreeofavertexu i, v j )ofg G is given by d G G u i, v j )n d G u i )+n d G v j ) d G u i )d G v j ), while the number of edges in G G is n m + n m 4m m Theorem 6 The first and second multiplicative Zagreb indices of the symmetric difference G G of two graphs G and G are bounded above as follows: i) G G ) n n n ) n n M G )+n M G )+4M G )M G ) +8n n m m 8n m M G ) 8n m M G ) ] n n, ) ii) G G ) M G )n +4M G ) 8n m )+M G )n 8n ] m )+8n n m m Q, 4) Q where Q u i V G ) v j V G ) P n m + n m 4m m ) and M G i ) is the first Zagreb index of G i, for i,moreover, the equalities in ) and 4) hold if and only if G G is a regular graph Proof We have d G G u i, v j )n d G u i )+n d G v j ) d G u i )d G v j ) By the definition of the first multiplicative Zagreb index, we have G G ) d G G u i, v j ) u i,v j ) V G G ) n d G u i )+n d G v j ) d G u i )d G v j ) ) u i V G ) v j V G )

13 Das et al Journal of Inequalities and Applications 0, 0:90 Page of 4 u i V G ) v j V G ) n d G u i )+n d G v j ) d G u i )d G v j )) ] n n 5) n n n n ) n n n M G )+n M G )+4M G )M G )+8n n m m 8n m M G ) 8n m M G ) ] n n The equality holds in 5) ifandonlyifd G u i )d G u k ), u i, u k VG )andd G v j ) d G v l ), v j, v l VG ) by Lemma ), that is, both G and G are regular graphs, which implies that G G is a regular graph By the definition of the second multiplicative Zagreb index, we have G G ) u i,v j )u k,v l ) EG G ) u i V G ) v j V G ) P P, d G G u i, v j )d G G u k, v l ) where P n d G u i )+n d G v j ) d G u i )d G v j ) Using the weighted arithmetic-geometric mean inequality in ), we get u i VG ) v j VG ) P P u i VG ) v j VG ) n d G u i )+n d G v j ) d G u i )d G v j )) ] u i V G ) v j V G ) P u i VG ) v j VG ) P 6) M G )n +4M G ) 8n m )+M G )n 8n ] m )+8n n m m Q, Q where Q u i V G ) v j V G ) P n m + n m 4m m ) First part of the proof is over The equality holds in 6) ifandonlyifd G u i )d G u k ), u i, u k VG )andd G v j ) d G v l ), v j, v l VG ) by Lemma ), that is, both G and G are regular graphs, which implies that G G is a regular graph Example 6 G G )p + q ) pq and G G )p + q ) pqp+q ) Competing interests The authors declare that they have no competing interests Authors contributions All authors completed the paper together All authors read and approved the final manuscript Author details Department of Mathematics, Sungkyunkwan University, Suwon, , Republic of Korea Department of Mathematics, Faculty of Arts and Science, Uludağ University, Gorukle Campus, Bursa, 6059, Turkey Department of Mathematics, Faculty of Science, Selçuk University, Campus, Konya, 4075, Turkey Acknowledgements Dedicated to Professor Hari M Srivastava All authors except the first one are partially supported by Research Project Offices of Uludağ 0-5and 0-9) and Selçuk Universities KC Das thanks for support the Sungkyunkwan University BK Project, BK Math Modeling HRD Div Sungkyunkwan University, Suwon, Republic of Korea Received: 9 December 0 Accepted: 5 February 0 Published: 5 March 0

14 Das et al Journal of Inequalities and Applications 0, 0:90 Page 4 of 4 References Todeschini, R, Ballabio, D, Consonni, V: Novel molecular descriptors based on functions of new vertex degrees In: Gutman, I, Furtula, B eds) Novel Molecular Structure Descriptors - Theory and Applications I, pp 7-00 Univ Kragujevac, Kragujevac 00) Todeschini, R, Consonni, V: New local vertex invariants and molecular descriptors based on functions of the vertex degrees MATCH Commun Math Comput Chem 64, ) Eliasi, M, Iranmanesh, A, Gutman, I: Multiplicative versions of first Zagreb index MATCH Commun Math Comput Chem 68, 7-0 0) 4 Gutman, I: Multiplicative Zagreb indices of trees Bull Soc Math Banja Luka 8, 7-0) 5 Liu, J, Zhang, Q: Sharp upper bounds on multiplicative Zagreb indices MATCH Commun Math Comput Chem 68, -40 0) 6 Xu, K, Hua, H: A unified approach to extremal multiplicative Zagreb indices for trees, unicyclic and bicyclic graphs MATCH Commun Math Comput Chem 68, ) 7 Xu, K, Das, KC: Trees, unicyclic and bicyclic graphs extremal with respect to multiplicative sum Zagreb index MATCH Commun Math Comput Chem 68), ) 8 Bondy, JA, Murty, USR: Graph Theory with Applications Macmillan Co, New York 976) 9 Khalifeh, MH, Azari, HY, Ashrafi, AR: The hyper-wiener index of graph operations Comput Math Appl 56, ) 0 Khalifeh, MH, Azari, HY, Ashrafi, AR: The first and second Zagreb indices on of some graph operations Discrete Appl Math 57, ) Imrich, W, Klavžar, S: Product Graphs: Structure and Recognition Wiley, New York 000) Harary, F: Graph Theory, p Addison-Wesley, Reading 994) doi:086/09-4x-0-90 Cite this article as: Das et al: The multiplicative Zagreb indices of graph operations Journal of Inequalities and Applications 0 0:90