Analysis of Some Bistar Related MMD Graphs

Transcription

1 Volume 118 No , ISSN: (printed version); ISSN: (on-line version) url: doi: /ijpam.v118i10.41 ijpam.eu Analysis of Some Bistar Related MMD Graphs S.Ganesh 1 and R.Revathi 2 1,2 Department of Mathematics, Sathyabama University,Chennai,India. sganesh19@yahoo.com, revathirangan75@gmail.com Abstract A graph G with n number of vertices is said to have modular multiplicative divisor (MMD) labeling if there exist a bijection function f from vertices of G to the set of all natural numbers less than n+1 with edge labels f(u)f(v) (mod n) for all u, v V (G) and uv E(G) such that n divides the sum of all edge labels of G. Here we discuss MMD labeling of a bistar related graphs. We prove that bistar related graphs such as shadow graph of bistard 2 (B n,n ), square graph of bistar are MMD graphs and studied the structural properties of MMD graphs. AMS Subject Classification: 05C78. Keywords: Graph labeling, Bistar Graph, Split Graph, Shadow Graph, Square Graph. 1 Introduction Throughout this paper all graphs under consideration are connected, undirected, finite and simple. We follow Harary [1] for terms and expressions in graph theory. A one-to-one mapping from set of graph elements on to a set of numbers is called graph labeling. Burton [2] identified that graph labeling is a strong implementation in number theory and also modular multiplication plays an important role in number theory problems. L.W.Beineke and S.M.Hedge [3] studied the application of number theory in structure 1 407

2 of graphs. Many results concerning labeling of graphs are summarized in the paper surveyed by Gallian [4]. Motivated by the result of H.T. Marbun [5], we [6] construct large families of graphs using the operations corona, union and addition. Also we [7] characterize certain families of MMD graphs. Here we discuss the existence of MMD labeling of bistar related graphs. 2 Basic Definitions Definition 2.1. A graph labeling is one-to-one mapping from set of graph elements on to a set of numbers subject to certain conditions. Definition 2.2. The graph constructed by joining the apex vertices of two stars K 1,r and K 1,s for r 1 and s 1 with disjoint vertex sets is known as bistar B r,s graph. Definition 2.3. Take two copies G and G of a connected graph and join each vertex v in G to the neighbours of the corresponding vertex v in G. This graph is known as shadow graph. Definition 2.4. A simple connected graph G is said to be a square graph G 2 if two adjacent vertices are at a distance 1 or 2 apart in G. 3 Main Results Application of number theory in graph coloring was discussed [8]. Throughout this section we consider the bistar B n,n with 2(n+1) vertices and 2n+1 edges. Theorems under this section discuss MMD labeling of shadow graph of B n,n square graph of B n,n and property of B r,s. Theorem 3.1. The shadow graph of bistar B n,n admits MMD labeling.. Proof. Let G be the shadow graph of B n,n with the vertex 5 set {v 1, v 2,.., v 2(n+1), v 1, v 2,..., v 2(n+1) } and the edge set E = 2 i=1 E i 408

3 where, E 1 = {v i v n+1 }, E 2 = {v i v n+2 }, E 3 = {v iv n+1}, E 4 = {v iv n+2} for 1 i n and n + 3 i 2n + 2, E 5 = {v n+1 v n+1uv n+1 v n+2uv n+2 v n+1uv n+2 v n+2} Allocate the labels to the vertices of G as follows. f(v n+1 ) = 1 f(v n+2 ) = 4n + 3 f(v n+1) = 2 f(v n+2) = 4n + 2 f(v 2n+2) { = 4(n + 1) 2+i 1 i n, f(v 1 ) = i n + 3 i 2n + 2 { 2n+2+i 1 i n, f(v 1) = 2n+i n + 3 i 2n + 2 For all v i v j E and v i, v j V the edge labels are f(v i )f(v j ) mod (4(n + 1)). Let S be the sum of labels of all the edges. Then S = 4(n + 1)(8n n + 6)(mod(4n+4))which is congruent to 0(mod(4n+4)). Hence the shadow graph of B n,n is a MMD graph. Illustration 3.2. MMD labeling of the shadow graph of bistar B 5,5 is shown in figure 1. Figure 1: MMD labeling of the shadow graph of B 5,

4 Theorem 3.3. labeling. The square graph of bistar B n,n admits MMD Proof: Let G be the square graph of B n,n with the vertex set {v 1, v 2,..., v 2(n+1) } and the edge set E 1 UE 2 where, E 1 = {v i v 2(n+1), 1 i 2n}, E 2 = {v i v 2(n+2), 1 i 2n + 1} Vertex labeling f : V {1, { 2,...2(n + 1)} is as follows i 1 i n f(v 2n+2 ) = n + 1, f(v i ) = i+1 n + 1 i 2n + 1 Edge labeling f : E(G) {0, 1, 2,..., 2n + 1} is f (v i v j ) =f(v i )f(v j ) (mod(2n+2)) for all v i v j E and v i, v j V. Then the sum of all labels of the edges is equal to 2(n + 1) 2 (1+3n)mod(2n+2) which is congruent to 0(mod(2n+2)). Hence the square graph of B n,n is an MMD graph. Illustration 3.4. MMD labeling of the square graph of B 5,5 is shown in figure 2. Figure 2: MMD labeling of the square graph of bistar B 5,

5 Theorem 3.5. for all r and s. Shadow graph of bistar B r,s is a MMD graph Let the bistar graph B r,s be G with {u, v, u 11,...u 1r, v 11,...v 1s } as the vertex set and u, v as the apex vertices of k 1,r, k 2,s respectively. Let the copy of G be G with {u, v, u 11,...u 1r, v 11,...v 1s} as the vertex set and the apex vertices of u, v as u, v respectively. Shadow graph of G is constructed by joining each vertices of G to the neighbours of the respective vertices of G. The number of vertices of a shadow graph is 2(r + s + 2) = N(say). Vertices of a shadow graph u 1i, u 1i, 1 i r are adjacent to u and u. Also v 1i, v 1i, 1 i s are adjacent to v and v. Assign f(u) = 1, f(u ) = N 1, f(v) = 2, f(v ) = N 2, we get the edge labels as f(u 1i )f(u) mod N,f(u 1i )f(u ) mod N for 1 i r, f(v 1j )f(v) mod N, f(v 1j )f(v ) mod N for 1 j s, f(u)f(v) mod N,f(u)f(v ) mod N,f(u )f(v ) mod N and f(v)f(u ) mod N. Let the sum of all labels of the edges be S. Then r f(u 1i ) [ f(u) + f(u ) ] s (modn) + f(v ji ) [ f(v) + f(v ) ] i=1 j=1 S= (modn) + f(u) [ f(v) + f(v ) ] (modn) + f(u ) [ f(v) + f(v ) ] (modn) [ r ] s S=N f(u 1i ) + f(v 1j )+N (mod N ) i=1 i=1 S 0(mod N ) Hence shadow graph of bistar B r,s is a modular multiplicative divisor graph for all r and s. 4 Concluding Remark We verify the existence of MMD labeling for bistar related graphs and also study the structural property of shadow graph of bistar. Our work under research is to investigate the maximum number of edges λ(n) in a MMD graph of order n > 3, except n 1 (mod2) and n 1,2 (mod3). Also to find for r s, MMD labeling of bistar B r,s

6 References [1] F. Harary, Graph Theory, Addison-Wesley, Reading, Massachusetts, (1972). [2] D.M. Burton, Elementary Number Theory, Second Edition, Wm. C. Brown Company Publishers, (1980). [3] L.W. Beineke and S.M. Hedge, Strongly Multiplicative Graph, Discuss, math. Graph Theory, 21 (2001), [4] R. Revathi and S.Ganesh, Characterization of Some Families of Modular Multiplicative Divisor Graphs, Journal of Taibah University for Science. Elsevier Sci Ltd, 11 (2017), [5] H.T. Marbun and A.N.M. Salman, Wheel-supermagic labelings for a wheel k-multilevel corona with a cycle, AKCE Int. J. Graphs Comb., 10,No 2 (2013), [6] R. Revathi and S.Ganesh, Modular Multiplicative Divisor Labeling of -Multilevel Corona Related Graphs, Journal of Computational and Theoretical Nanoscience, 13 (2016), [7] J. A. Galliani, A Dynamic Survey of Graph Labeling, The Electronic journal of combinatorics, (2010). [8] R. Mary Jeya Jothi, Domination and Co - domination Numbers of Cube Connected Cycles, Applied Mathematical Sciences,9 No 14 (2015),

7 413

8 414