Some Graph Operations Of Even Vertex Odd Mean Labeling Graphs

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1 Some Graph Operations Of Even Vertex Odd Mean Labeling Graphs M.Kannan, R.Virama Prasad and R.Gopi 3 Research Scholar,Research & Development Centre, Bharathiar University, Coimbatore , India. Department of Mathematics, Government Arts College(Autonomous), Salem , India. 3 PG and Research Department of Mathematics(Autonomous), Srimad Andavan Arts and Science College, Tiruchirappalli 60005, Tamil Nadu, India. *(corresponding author) Orcid: Abstract A graph with p vertices and q edges is said to have an even vertex odd mean labeling if there exists an injective function f:v(g) {0,, 4,... q-,q} such that the induced map f*: E(G) {, 3, 5,... q-} defined by f*(uv)= f u f v is a bijection. A graph that admits an even vertex odd mean labeling is called an even vertex odd mean graph. In this paper we pay our attention to prove some graph operations of even vertex odd mean labeling graphs Keywords: Even vertex odd mean labeling, even vertex odd mean graph AMS subject classification (00): 05C78 INTRODUCTION Throughout this paper we restrict our attention to finite, simple and undirected graphs.the set of vertices and the set of edges of a graph G will be denoted by V(G) and E(G) respectively and let p = V(G), q = E(G).For general graph theorectic notations we follow F.Harary[7].A graph labeling is a mapping that carries a set of elements(usually vertices and /or edges) into a set of numbers.many inds of labeling have been studied an excellent survey of graph labeling can be found in[].most of the graph labeling techniques found their origin with graceful labeling which was introduced by Rosa.A(967). Let G(V,E) be a graph with p vertices and q edges. The concept of mean labeling was introduced and studied by Somasundaram and Ponraj []. Further some more results on mean graphs are discussed in [4,6]. A graph G is said to be a mean graph if there exists an injective function f: V(G) {0,,, q} such that the induced map f*: E(G) {,, q} defined by f* (uv)= f u f v bijection. is a Manicam and Marudai [0] have introduced the concept of odd mean labeling of a graph. A graph G is said to be odd mean if there exists an injective map f:v(g) {0,, q } is a bijection. The concept of even mean labeling was introduced and studied by Gayathri and Gopi [3]. A graph G is said to be even mean if there exists an injective function defined by f*(uv) = f u f v f: V(G) {0,, q} such that the induced map f*:e(g) {,4, q} defined by f*(uv) = f u f v bijection. is a A graph G is said to have an even vertex odd mean labeling if there exists an injective function f:v(g) {0,, q, q} such that the induced map f*:e(g) {, 3, q } defined f by f*(uv) = u f v is a bijection. A graph that admits an even vertex odd mean labeling is called even vertex odd mean graph [, 6]. In this paper, we proved that the even vertex odd meanness of some graph operations of even vertex odd mean labeling graphs Definition :.[8] The shadow graph D (G) of a connected graph G is constructed by taing two copies of G say G and G". Join each vertex u in G to the neighbours of the corresponding vertex v in G". 7749

2 Definition :.[5] Consider a cycle C n and let e = v v + be an edge in it with e = v v and e + = v + v + be its incident edges and e = v v be a new edge. The duplication of an edge e by an edge e produces a new graph G in such a way that N v N v v and N v N v v which is called edge duplication of C n and denoted by ED C where n adjacent to v N v denotes the set of vertices Definition :.3[4] For a bipartite graph with G with partite sets v and v. Let G be the copy of G and v and v be the copies of v and v. The mirror graph M(G) of g is obtained from G and G by joining v to its corresponding vertex in v by an edge. Definition :.4[3] Le P m + be K n the graph with the vertex set V (G) ={u i,v j : i m, j n} and the edge set E(G) = {u iu i+,u v j,u mv j : i m, j n}. Definition :.5[7] Let G = (V,E) be a simple graph and G 0 = (V 0,E 0 ) be another copy of graph G. Join each vertex v of G to the corresponding vertex v 0 of G 0 by an edge.the new graph thus obtained we call - tuple graph of G. We denote -tuple graph of G by the notation T (G). Further we note that if G = (p,q) then V(T (G)) = p and (T (G)) = q + p Figure. : Ordinary labeling of D (, n) Define f : v {0,, q} by Let For i n f(v) = 0 and f(u) = 8n f(v i) = 4i- f(u i) = 4n+4i- For i 4n f*(a i) = i Therefore f*(e) = {, 3, 5,...q }. So, f is an even vertex odd mean labeling and hence, the graph D (, n), n is an even vertex odd mean graph.even vertex odd mean labeling of D (, 4) is shown in Figure. : Definition :.6[5] T t,n,m is a graph obtained by joining the centers of K,n and K,m by a path P t. It consists of t+n +m vertices and t+n +m - edges. MAIN RESULTS Theorem:. The graph D (, n), n is an even vertex odd mean graph. Proof : Let {v,v i, i n, u,u i, i n} be the vertices and {a i, i 4n} be the edges which are denoted as in figure. Figure. : Even vertex odd mean labeling of D (, 4) 7750

3 Theorem :. The mirror graph M(P n ),n 3 is an even vertex odd mean graph. Proof: Let { v i,v i, i n} be the vertices and {e i,e i i n-,, i n} be the edges which are denoted as in figure.3 a i Therefore f*(e) = {, 3, 5,...q }. So, f is an even vertex odd mean labeling and hence the graph M(P n ), n 3 is an even vertex odd mean graph.the below Figure.4 provides the better idea about the above defined mean labeling a n Figure.4: Even vertex odd mean labeling of M(P 6 ) Figure.3: Ordinary labeling of M(P n ) Define f : v {0,, q} by For i n f(v i) = 6(i-) For i n f(v i ) = 6i-4 Theorem.3 The graph D (B n,n), n 3 is an even vertex odd mean graph Proof : Let {v,v i i n,u,u i, i n,x,x i, i n,w,w i, i n } be the vertices and {a i, i n, a i, i n, a, a, a, b i i n, b i,, i n } be the edges which are denoted as in figure.5 For i n f*(a i) = 6i 5 For i n - f*(e i) = 6i 3 For i n- f*(e i ) = 6i Figure.5: Ordinary labeling of D (B n,n) 775

4 Define f : v {0,, q} by f(u)= 8n+6 For i n- f(u i) = 4n+4i, f(u n) = 8n+4 For i n f(v i) = 8n+4(i +), f(v) = n+6 f(w) = 4n, f(w ) = 4n+ For i n f(w i) = 4n+4i+ f(x) =0 For i n f(x i) = 4i- Theorem:.4: The graph graph Proof: T (P n P ), n 3 is an even vertex odd mean Let { v i,u i, u i,, v i, i n} be the vertices {a i, a i,, bi, b i, i n, c i, c i,, i n} d i i n } be the edges which are denoted as in figure.7 For i n- f*(a i ) = 6n+i+3 f*(a n ) = 8n+5 For i n f*(a i) = 8n+i+5 f*(a) = 4n+3, f*( a) = 8n+3, f*( a) = 6n+3,f*(b )) = 4n+ For i n f*(b i ) = 4n+i+ Fo i n f*(b i) = i Therefore f*(e) = {, 3, 5,...q }. So, f is an even vertex odd mean labeling and hence the graph D (B n,n),n 3is an even vertex odd mean graph. Even vertex odd mean labeling of D (B 5,5) is shown in figure.6 Figure.7: Ordinary labeling of T (P n P ) Define f : v {0,, q} by For i n f(u i) = (i-), f(u i )) = 0(n-)+6i For i n f(v i) = 4n+i-4, f(v i ) = 0n+6i-8 For i n- f*(a i) = i- For i n f*(b i) = n+i-3 For i n- f*(c i) = 4n+i-3 For i n f*(d i) = 6n+i-5 For i n- f(a i )) = 0n+6i-7 For i n f(b i )) = 0n+6i-9 For i n- f(c i )) = 0n+6i-5 Figure.6: Even vertex odd mean labeling of D (B 5,5) Therefore f*(e) = {, 3, 5,...q }. So, f is a even vertex odd mean labeling and hence the graph T (P n P ), n 3 is an even vertex odd mean graph. The below illustration provides the better undestanding of defined labeling pattern in the above theorem 775

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