International Journal of Mathematics Trends and Technology- Volume3 Issue1- 2012 On Strongly Multiplicative Graphs *M. Muthusamy 1, K.C. Raajasekar 2, J. Baskar Babujee 3 1SCSVMV University, Enathur, Kanchipuram – 631 561, India. 2Department of Mathematics, Presidency College, Triplicane, Chennai - 600 005, India. 3Department of Mathematics, Anna University, Chennai-600 025, India. Abstract — A graph G with p vertices and q edges is said to be an induced subgraph of a strongly multiplicative graph. In this strongly multiplicative if the vertices are assigned distinct paper we study strongly multiplicative labeling for some numbers 1, 2, 3, …, p such that the labels induced on the edges special classes of graphs. by the product of the end vertices are distinct. We prove some of the special graphs obtained through graph operations such as + Definition 1: Cn (a graph obtained by adding pendent edge for each vertex of d A graph G = (V(G), E(G)) with p vertices is said to the cycle Cn), (Pn mK1) +N2, Pn + mK1 and Cn (cycle Cn with non-intersecting chords) are strongly multiplicative. be multiplicative if the vertices of G can be labeled with distinct positive integers such that label induced on the edges AMS Subject Classification: 05C78 by the product of labels of end vertices are all distinct. Keywords— Graph, labeling, path, cycle, relation, function. Definition 2: A graph G = (V(G), E(G)) with p vertices is said to I. INTRODUCTION be strongly multiplicative if the vertices of G can be labeled with p distinct integers 1, 2 ,..., p such that label induced on In this paper we deal with only finite, simple, connected the edges by the product of labels of the end vertices are all and undirected graphs obtained through graph operations. A distinct. labeling of a graph G is an assignment of labels to vertices or edges or both following certain rules. Labeling of graphs plays II. MAIN RESULTS an important role in application of graph theory in Neural Theorem 1: Networks, Coding theory, Circuit Analysis etc. A useful survey on graph labeling by J.A. Gallian (2010) can be found The graph Cn has strongly multiplicative labeling. in [5]. All graphs considered here are finite, simple and Proof : Let V v , v , v ,......v , v ,.....v be the vertex undirected. In most applications labels are positive (or 1 2 3 n n1 2n nonnegative) integers, though in general real numbers could set and E E1 E2 E3 be the edge set where be used. Beineke and Hegde [4] call a graph with p vertices strongly E1 vivi1, 1 i n, E2 vivni , 1 i n and multiplicative if the vertices of G can be labeled with distinct E v v of the graph C . Define a bijection integers 1, 2, . , p such that the labels induced on the edges 3 n 1 n by the product of the end vertices are distinct. They prove the f :V 1,2,3,.....2n such that following graphs are strongly multiplicative: trees; cycles; Case (i): When n is even wheels; Kn if and only if n ≤ 5; Kr,r if and only if r ≤ 4; and n Pm× Pn. Beineke and Hegde [4] obtain an upper bound for the 4i 3 1 i maximum number of edges λ(n) for a given strongly f v 2 i n multiplicative graph of order n. It was further improved by 4n i 3 1 i n C. Adiga, H. N. Ramaswamy, and D. D. Somashekara [2] for 2 greater values of n. It remains an open problem to find a and nontrivial lower bound for λ(n). Seoud and Zid [7] prove the following graphs are strongly multiplicative: wheels; rK for n n 4i 2 1 i all r and n at most 5; rKn for r ≥ 2 and n = 6 or 7; rKn for r ≥ 3 2 f vni and n = 8 or 9; K for all r; and the corona of P and K for n 4,r n m 4n i 4 1 i n all n and 2 ≤ m ≤ 8.Germina and Ajitha [6] prove that K2 + Kt, 2 quadrilateral snakes, Petersen graphs, ladders, and unicyclic Define an induced function g : E N, such that graphs are strongly multiplicative. Acharya, Germina, and Ajitha [1] have shown that every graph can be embedded as gviv j f vi f v j viv j E and vi , v j V We show that the labeling of edges within the edge sets and * Department of Mathematics, among the edge sets are distinct. If it is assumed in each case Rajalakshmi Engineering College, Thandalam–602105, India. ISSN: 2231-5373 http://www.internationaljournalssrg.org Page 13 International Journal of Mathematics Trends and Technology- Volume3 Issue1- 2012 that the induced label of the edges are same then we arrive at 8n 7 contradiction. gives i = j a contradiction as i j is not possible. 4 n n For the edges in E1 : Sub Case (c) For 1 i and 1 j n also i j n 2 2 Sub Case (a) For i j and 1 i, j in E : 2 2 2 If we assume that We see that gv v 16i 20i 6 gvivi1 gv jv j1 i ni g v v 16 j 2 20 j 6 16n2 32nj 28n 8 j 12 f vi f vi1 f v j f v j1 j n j 4i 3 4i 1 3 4 j 3 4 j 1 3 gvivni gv jvn j 1 Clearly the labeling of edges of E and that of E are all gives i = j a contradiction as i j is not possible. 1 2 2 distinct as the labeling of edges of E1 are all odd and those of n are even. Also edges of & and & are also Sub Case (b) For i j and 1 i, j n E2 E1 E3 E2 E3 2 distinct as edge in E3 is with the minimum odd label 3. If we assume that gvivi1 gv jv j1 Cn has strongly multiplicative labeling for n even. f vi f vi1 f v j f v j1 4n i 34n i 1 3 4n j 34n j 1 3 4n 1 gives i = j a contradiction as i j is not possible. 2 n n Sub Case (c) For 1 i and 1 i n also i j 2 2 in E1 2 We have g v i v i 1 16 i 8 i and g v v 16 j 2 8 j 16 n 2 8n 32 nj j j1 Clearly 16i2 8i 16 j 2 8 j 16n2 8n 32nj gvivi1 gv jv j1 For the edges in E2 : n Fig. 1. Strongly multiplicative labeling for C Sub Case (a) For i j and 1 i, j 8 2 Case (ii): When n is odd If we assume that gvivni gv jvn j The bijection f : V 1,2,3,.....2nis defined as the f v f v f v f v i ni j ni following 4i 34i 2 4 j 34 j 2 n 1 4i 3 1 i 5 2 gives i = j a contradiction as is not possible. f v i j i n 1 4 4n i 3 1 i n n 2 Sub Case (b) For i j and 1 i, j n and 2 n 1 If we assume that gvivni gv jvn j 4i 2 1 i 2 f vni f vi f vni f v j f vn j n 1 4n i 4 1 i n 4n i 34n i 4 4n j 34n j 4 2 ISSN: 2231-5373 http://www.internationaljournalssrg.org Page 14 International Journal of Mathematics Trends and Technology- Volume3 Issue1- 2012 n f v f v f v f v The proof follows as above, replacing 1 i by i n2 j n2 2 in 2 jn 2 n 1 n n 1 i j a contradiction. 1 i and 1 i n by 1 i n . 2 2 2 For the edges in E2 : Thus Cn has strongly multiplicative labeling for n odd. For i j, 1 i, j n 3 Hence Cn has strongly multiplicative labeling for all n. If we assume gvivn1 gv jvn1 f vi f vn1 f v j f vn1 in 1 jn 1 i j a contradiction For the edges in E3 : For i j, 1 i, j 3 If we assume gvnvni gvnvn j f vn f vni f vn f vn j nn i nn i i j a contradiction Now we have to show that the edges between different edge sets are distinct. For the edges in E1 and E2 : Fig.2. strongly multiplicative labeling for C9 For i j, 1 i, j n 3 Theorem 2: The graph P mk N has strongly 2 1 2 If we assume gvivn2 gv jvn1 multiplicative labeling. f vi f vn2 f v j f vn1 Proof: Let V v1, v2 , v3,......vm ,vm1,.....vm4, where in 2 jn 1 n m 4 be the vertex set and E E1 E2 E3 be the 2i j edge set where E v v , 1 i n 3, i j a contradiction 1 i n2 n E2 vi vn1 , 1 i n 3 and For the edges in E and E : E3 vnvn1,vnvn2 ,vnvn3 of the graph P2 mk1 N2 . 1 3 For Define a bijection f :V 1,2,3,.....n such that i j, 1 i n 3, j 1,2,3 If we assume gvivn2 gvnvn j f vi i, 1 i n Define an induced function g : E N, such that f vi f vn2 f vn f vn j gv v f v f v v v E and v , v V n 2 n j i j i j i j i j a contradiction We show that the labeling of edges within the edge sets and n i among the edge sets are distinct.