The Multiplicative Zagreb Indices of Products of Graphs

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1 Iteratioal Joural of Mathematics Research. ISSN Volume 8, Number (06), pp Iteratioal Research Publicatio House The Multiplicative Zagreb Idices of Products of Graphs R. Murugaadam, R. S. Maikada ad Aruvi M. 3 Departmet of Mathematics, Govermet Arts College, Trichy, Tamil Nadu, Idia. Departmet of Mathematics, Bharathidasa Costituet College, Lalgudi, Trichy, Tamil Nadu, Idia. 3 Departmet of Mathematics, Aa Uiversity BIT Campus, Trichy, Tamil Nadu, Idia. Abstract I this paper, we determie the exact formulae for the multiplicative Zagreb idices of strog ad tesor products of two coected graphs. Also we apply our results to compute the multiplicative Zagreb idices of ope ad closed fece graphs. Keywords: The multiplicative Zagreb idices, strog product ad tesor product.. Itroductio Throughout this paper, we cosider simple graphs which are fiite, idirected graphs without loops ad multiple edges. Suppose G is a graph with vertex set V(G) ad a edge set E(G). For a graph G, the degree of a vertex v is the umber of edges icidet to v ad is deoted by d G (v). A topological idex Top(G) of a graph G is a umber with the property that for every graph H is isomorphic to G, Top(H) = Top(G). The Zagreb idices have bee itroduced more tha thirty years ago by Gutma ad Friajestic [. They are defied as M (G) = u V(G) (d G (u)) = uv E(G) [d G (u) + d G (v) ad M (G) = [d G (u)d G (v) uv E(G).

2 6 R. Murugaadam, R.S.Maikada ad Aruvi M The Zagreb idices are foud to have applicatios i QSPR ad QSAR studies as well see [3. Recetly, Todeschii et al. [4, 5 have proposed the multiplicative variats of ordiary Zagreb idices, which are defied as follows: = (G) = [d G (u) = [d G (u) + d G (v) u V(G) uv V(G) ad = (G) = uv E(G) [d G (u)d G (v) Mathematical properties ad applicatios of multiplicative Zagreb idices are reported i [4-9.Mathematical properties ad applicatios of multiplicative sum Zagreb idices are reported i [0. I [, K.Pattabirama computed Zagreb idices ad coidices of product graphs. The Strog product of graphs G ad G,deoted by G G, is the graph with vertex set V(G ) V(G ) = {(u, v): u V(G ), v V(G )} ad (u, x)(v, y) is a edge wheever (i) u = v ad xy E(G ) or (ii)uv E(G ) ad x = y or (iii) uv E(G ) ad xy E(G ). For two simple graphs G ad G their tesor product, deoted by G G, has the vertex set V(G ) V(G ), i which (u, v) ad (x, y) are adjacet wheever ux is a egde i G ad vy is a edge i G. Note that if G ad G are coected graphs, the G X G is coected oly if atleast oe of the graph is obipartite. I this paper, we compute the multiplicative Zagreb idices of strog ad tesor products of two coected graphs. Moreover, computatios are doe for some well kow graphs.. The Multiplicative Zagreb Idices of G G We begi this sectio with stadard iequality as follows: Lemma. (Arithmetic Geometric Iequality) Let x, x,, x be o-egative umbers. The x +x + +x x x.. x holds with equality if ad oly if all x k s are equal. I this sectio, we compute the multiplicative Zagreb idices of the strog product of graphs. The followig lemma follows from the defiitio of the strog product of the two graphs. Lemma. Let G ad G be two graphs. (i). V(G G ) = V(G ) V(G ) (ii). E(G G ) = V(G ) E(G ) + V(G ) E(G ) + E(G ) E(G ) (iii). d G G (u i, u j ) = d G (u i ) + d G (v j ) + d G (u i )d G (v j ).

3 The Multiplicative Zagreb Idices of Products of Graphs 63 Theorem.3 Suppose G ad G are graphs with V(G ) =, V(G ) =, E(G ) = m ad E(G ) = m. The (G G ) [ ( +4m )M (G )+( +4m )M (G )+M (G )M (G )+8m m where M (G ) is the first Zagreb idex of G. Proof: By the defiitio of the multiplicative first Zagreb idex, (G G ) = [d G G (u i, v j ) (u i,v j ) V(G G ) = [d G (u i ) + d G (v j ) + d G (u i )d G (v j ) u i V(G ) v j V(G ) by lemma. [ [ dg (u i )+d G (v j )+d G (u i )d G (v j )+d G (u i )d G (v j) u v +d G (u i )d G (v j)+d G (u i )d G i V(G ) j V(G ) (v j ) by lemma. (G G ) [ ( +4m )M (G )+( +4m )M (G )+M (G )M (G )+8m m by the defiitio of the first Zagreb idex of G. I G G, defie E = {(u, v)(x, y) E(G G ) ux E(G ) ad v = y}, E = {(u, v)(x, y) E(G G ) ux E(G ) ad vy E(G )}, E 3 = {(u, v)(x, y) E(G G ) u = x ad vy E(G )}. Clearly, E E E 3 = E(G G ). Also E = E(G ) V(G ), E = E(G ) E(G ) ad E 3 = E(G ) V(G ). Theorem.4 Let G ad G be two graphs with ad vertices, m ad m edges respectively. The (G G ) (m ) m [ M (G ) + m M (G ) + m M (G ) + 4m M (G ) + M (G )M (G ) + M (G )M (G ) m X

4 64 R. Murugaadam, R.S.Maikada ad Aruvi M (m m ) m m [m M (G ) + m M (G ) + M (G )M (G ) + M (G )M (G ) + M (G )M (G ) + 8m m m m X (m ) m [m M (G ) + m M (G ) + M (G ) + 4m M (G ) + M (G )M (G ) + M (G )M (G ) m where M (G) ad M (G) are the first ad secod Zagreb idices respectively. Proof: From the above partitio of the edge set i G G, we have G G = d G G (u i, v j )d G G (u p, v q ) (u i,v j )(u p,v q ) E(G G ) = d G G (u i, v j )d G G (u p, v j ) (u i,v j )(u p,v j ) E (u i,v j )(u p,v q ) E d G G (u i, v j )d G G (u p, v q ) X d G G (u i, v j )d G G (u i, v q ) (u i,v j )(u i,v q ) E 3 = A X B X C (.) X where A, B ad C are the products of the terms of the above expressios, i order. Now we calculate = [ u i u p E(G ) A = d G G (u i, v j )d G G (u p, v j ) (u i,v j )(u p,v j ) E v j V(G ) [d G (u i ) + d G (v j ) + d G (u i )d G (v j ) [d G (u p ) + d G (v j ) + d G (u p )d G (v j ) [ d G (u i )d G (u p)+m [d G (u i )+d G (u p)+m (G )+4m d G (u i )d G (u p)+ u M (G )[d G (u i )+d G (u p)+m (G )d G (u i )d G (u i up E(G ) p) by the defiitio of the first Zagreb idex. A [ (m ) m M (G ) + m M (G ) + m M (G ) + 4m M (G ) + M (G )M (G ) + M (G )M (G ) m (.)

5 The Multiplicative Zagreb Idices of Products of Graphs 65 by the defiitio of the secod Zagreb idex. Next we calculate = u i u p E(G ) B = d G G (u i, v j )d G G (u p, v q ) (u i,v j )(u p,v q ) E v j v q E(G ) [d G (u i ) + d G (v j ) + d G (u i )d G (v j ) [d G (u p ) + d G (v q ) + d G (u p )d G (v q ) d G (u i )d G (u p)+d G (u i )d G (v q)+d G (u i )d G (u p)d G (v q)+ u [ d G (u p)d G (v j)+d G (v j)d G (v q)+d G (u p)d G (v j)d G (v q) i u p E(G ) v j vq E(G ) +d G (u i )d G (u p)d G (v j)+d G (u i )d G (v j)d G (v q) m m m m B [ (m m ) m m [m M (G ) + m M (G ) + M (G )M (G ) + M (G )M (G ) + M (G )M (G ) + 8m m m m by the defiitios of the first ad secod Zagreb idices of G. Fially we compute, C = d G G (u i, v j )d G G (u i, v q ) (u i,v j )(u i,v q ) E 3 [ = [d G (u i ) + d G (v j ) + d G (u i )d G (v j ) [d G (u i ) + d G (v q ) + d G (u i )d G (v q ) u i V(G ) v j v q E(G ) [ d G (u i )+d G (u i )[d G (v j)+d G (v q)+d G (u i )[d G (v j)+d G (v m q)+ u i V(G ) v d G (v j)d G (v q)+d G (u i )d G (v j)d G (v q)+d G (u i )d G (v j)d G (v q) j vq E(G ) m C (m ) m [m M (G ) + m M (G ) + M (G ) + 4m M (G ) + M (G )M (G ) + M (G )M (G ) m (.4)

6 66 R. Murugaadam, R.S.Maikada ad Aruvi M by the defiitios of the first ad secod Zagreb idices of G. Usig (.), (.3) ad (.4) i (.), we get (G G ) (m ) m [ M (G ) + m M (G ) + m M (G ) + 4m M (G ) + M (G )M (G ) + M (G )M (G ) m X (m m ) m m [m M (G ) + m M (G ) + M (G )M (G ) + M (G )M (G ) + M (G )M (G ) + 8m m m m X (m ) m [m M (G ) + m M (G ) + M (G ) + 4m M (G ) + M (G )M (G ) + M (G )M (G ) m By direct calculatios, we obtai the followig expressios as lemma: Lemma.4 Let P ad C deote the path ad cycle o vertices, respectively. (). For, M (P ) = 4 6 ad M (P ) = 0 (). For 3, M (C ) = 4, M (C ) = 4 ad M (P ) = 4( ) (3). For 3, M (K ) = ( ) ad M (K ) = ( )3 By usig Theorem.3, ad Lemma.4, we obtai the exact multiplicative Zagreb idices of ope ad closed feces P K ad C K, see Fig.. (i). (P K ) [ (ii). (C K ) [ + 4 +

7 The Multiplicative Zagreb Idices of Products of Graphs 67 (iii). (P K ) [50 (( )) ( ) 90( ) X 47 ( ) X [5 34 ( ) ( ) [9 (iv). (C K ) () [50 X [30 X [5 Figure.. Closed ad Ope fece graphs 3. The Multiplicative Zagreb Idices of G XG I this sectio, we compute the multiplicative Zagreb idices of the strog product of graphs. The followig lemma follows from the structure of G XG. Lemma 3. Let G ad G be two graphs. The (i). V(G XG ) = V(G ) V(G ) (ii). E(G XG ) = E(G ) E(G )

8 68 R. Murugaadam, R.S.Maikada ad Aruvi M (iii). The degree of a vertex (u i, v j ) of G XG is give by d G XG (u i, v j ) = d G (u i )d G (v j ). Theorem 3. Let G ad G be two graphs. The (G XG ) = (G ) (G ), where (G ) ad (G ) are the first multiplicative Zagreb idex of G ad G respectively. Proof: By the defiitio of the first multiplicative Zagreb idex, (G XG ) = [d (G XG )(u i, v j ) (u i,v j ) V(G XG ) = [d G (u i ) [d G (v j ) u i V(G ) v j V(G ) = (G ) (G ) by the defiitio of the first multiplicative Zagreb idex of the graphs G ad G respectively. Theorem 3.3 Let G ad G be two graphs. The (G XG ) = (G ) (G ), where (G ) ad (G ) are the secod multiplicative Zagreb idex of G ad G respectively. Proof: By the defiitio of the seod multiplicative Zagreb idex, (G XG ) = [d (G XG )(u i, v j )[d (G XG )(u p, v q ) (u i,v j )(u p,v q ) E(G XG ) = d G (u i ) d G (v j )d G (u p )d G (v q ) u i u p E(G ) v j v q E(G )

9 The Multiplicative Zagreb Idices of Products of Graphs 69 = d G (u i )d G (u p ) d G (v j )d G (v q ) u i u p E(G ) v j v q E(G ) = (G ) (G ) by the defiitio of the secod multiplicative Zagreb idex of the graphs G ad G respectively. Refereces [ R.Balakrisha ad K.Ragaatha, A Text Book of Graph Theory, Spriger- Verlog, New York, 000. [ I.Gutma, N.Triajstic, Graph Theory ad molecular orbits. Total π -electio eergy of alterate hydrocarbos, chem, Phy. Lett. 7(97); [3 J.Devillers, A.T.Balaba,Eds., Topological idices ad related descriptors i QSAR ad QSPR, Gordo ad Breach, Amstersam, The Netherlads, 999. [4 R.Todeschii, D.Ballabio, V.Cosoi: Novel molecular descriptors based o fuctios of ew vertex degrees. I: Gutma.I, Furtula.B (eds)novel molecular structure descriptors - Theory ad Applicatios I, pp Uiv. kragujeva, Kragujevac (00). [5 R.Todeschii, V.Cosoi, New local vertex ivariats ad molecular descriptors based o fuctios of the vertex degrees. MATCH Commu. Math. Comput. Chem. 64, (00). [6 M.Eliasi, A.Iramaesh, I.Gutma, Multiplicative versios of first Zagreb idex. MATCH Commu. Math. Comput. Chem. 68, 7-30(0). [7 I.Gutma, Multiplicative Zagreb idices of trees. Bull. Soc. Math. Baja Luka 8, 7-3(0). [8 J.Liu, Q.Zhag: Sharp upper bouds o multiplicative Zagreb idices. MATCH Commu. Math. Com-put. Chem. 68, 3-40(0). [9 Xu.K,Hua.H, A ui_ed approach to extremal multiplicative Zagreb idices for trees, uicyclic ad bicyclic graphs. MATCH Commu. Math. Comput. Chem. 68, 4-56(0). [0 Xu,K.Das,KC, Trees, uicyclic ad bicyclic graphs extremal with respect to multiplicative sum Zagreb idex. MATCH Commu. Math. Comput. Chem. 68(), 57-7(0). [ K.Pattabirama, S.Nagaraja, M.Chedrasekhara, Zagreb idices ad coidices of product graphs, Joural of Prime Research i Mathematics Vol 0, 80-9(05).