Skolem Difference Mean Graphs

Transcription

1 royecciones Journal of Mathematics Vol. 34, N o 3, pp , September 015. Universidad Católica del Norte Antofagasta - Chile Skolem Difference Mean Graphs M. Selvi D. Ramya Dr. Sivanthi Aditanar College of Engineering, India and. Jeyanthi Govindammal Aditanar College for Women, India Received : April 015. Accepted : June 015 Abstract A graph G =(V,E) with p vertices and q edges is said to have skolem difference mean labeling if it is possible to label the vertices x V with distinct elements f(x) from 1,, 3,,p+ q insuchaway l f(u) f(v) m that for each edge e = uv, letf (e) = and the resulting labels of the edges are distinct and are from 1,, 3,,q. A graph that admits a skolem difference mean labeling is called a skolem differencemeangraph. Inthispaper,weproveC m (n 3,m 1), T hk 1,n1 : K 1,n : : K 1,nm i, T hk 1,n1 K 1,n K 1,nm i, St(n 1,n,,n m ) and Bt(n, n,,n) are skolem difference mean graphs. Keywords : Mean labeling, skolem difference mean labeling, skolem difference mean graph, extra skolem difference mean labeling. AMS Subject Classification : 05C78

2 44 M. Selvi, D. Ramya and. Jeyanthi 1. Introduction Throughout this paper by a graph we mean a finite, simple and undirected one. The vertex set and the edge set of a graph G are denoted by V (G) and E(G) respectively. Terms and notations not defined here are used in the sense of Harary[1]. A graph labeling is an assignment of integers to the vertices or edges or both, subject to certain conditions. There are several types of labeling. An excellent survey of graph labeling is available in []. The concept of mean labeling was introduced by Somasundaram and onraj [8]. A graph G =(V,E)withp vertices and q edges is called a mean graph if there is an injective function f that maps V (G) to{0, 1,,,q} such that each edge uv is labeled with f(u)+f(v) if f(u)+f(v) isevenand f(u)+f(v)+1 if f(u)+f(v) is odd. Then the resulting edge labels are distinct. The notion of skolem difference mean labeling was due to Murugan and Subramanian [3]. A graph G =(V,E) withp vertices and q edgesissaidtohaveskolem difference mean labeling if it is possible to label the vertices x V with distinct elements f(x) l from1,, m3,,p+q in such a way that for each edge e = uv, letf (e) = f(u) f(v) and the resulting labels of the edges are distinct and are from 1,, 3,,q. A graph that admits a skolem difference mean labeling is called a skolem difference mean graph. Further studies on skolem difference mean labeling are available in [4]-[7]. In this paper, we extend the study on skolem difference mean labeling and prove that C m (n 3,m 1), T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i, T hk 1,n1 K 1,n K 1,nm i, St(n 1,n,,n m )andbt(n, n,,n)are skolem difference mean graphs. We use the following definitions in the subsequent section. Definition 1.1. The graph C m is obtained by identifying one pendant vertex of the path m with a vertex of the cycle C n. Definition 1.. The shrub St(n 1,n,,n m ) is a graph obtained by connecting a vertex v 0 to the central vertex of each of m number of stars. Definition 1.3. The banana tree Bt(n 1,n,,n {z m ) is a graph obtained } by connecting a vertex v 0 tooneleafofeachofm number of stars. Definition 1.4. Let G =(V,E) be a skolem difference mean graph with p vertices and q edges. If one of the skolem difference mean labeling of G

3 Skolem Difference Mean Graphs 45 satisfies the condition that all the labels of the vertices are odd, then we call this skolem difference mean labeling an extra skolem difference mean labeling and call the graph G an extra skolem difference mean graph [7]. An extra skolem difference mean labeling of 6 isgiveninfigure1. Main Results Theorem.1. The graph C m (n 3,m 1) is a skolem difference mean graph. roof. Case (i): n is odd. Let n =k +1. Let u 1,u,,u k,v k,v k 1,,v 1,v 0 be the vertices of C k+1 and let w 1,w,,w m be the vertices of m. Then C m is obtained by identifying w 1 of m with v 0 of C k+1, which has n + m 1 edges. E(C m )={v i v i+1 1 i k 1} {u i u i+1 1 i k 1} {w j w j+1 1 j m 1} {v 0 v 1,v 0 u 1,u k v k }.Define f : V (C m ) {1,, 3,,p+ q =n +m } as follows: f(w i 1 )=i 1for1 i m, f(w i )=m +3 i for 1 i m, l m f(u i 1 )=m +n + 4i for 1 i k, j k f(u i )=4i for 1 i k, l m f(v i 1 )=m +n +1 4i for 1 i k, j k f(v i )=4i +for1 i k. For each vertex label f, the induced edge label f is calculated as follows: f (w i w i+1 )=m +1 i for 1 i m 1, f (u k v k )=1, f (u i u i+1 )=n + m 1 i for 1 i k 1,

4 46 M. Selvi, D. Ramya and. Jeyanthi f (v i v i+1 )=n + m i for 1 i k 1, f (u 1 v 0 )=n + m 1, f (v 1 v 0 )=n + m. Case (ii): n is even. Let n =k, k>1. Let v 0,v 1,v,,v k 1,u 0,u k 1,u k,,u,u 1 be the vertices of C k and let w 1,w,,w m be the vertices of m.thenc m is obtained by identifying w 1 of m with v 0 of C k, which has n + m 1 edges. E(C m ) = {v i v i+1 1 i k } {u i u i+1 1 i k } {w j w j+1 1 j m 1} {v 0 v 1,v 0 u 1,u 0 u k 1,u 0 v k 1 }. Subcase (i): k>1 is odd. Define f : V (C m ) {1,, 3,,p+ q =n +m } as follows: f(w i 1 )=i 1for1 i m, f(w i )=m +1 i for 1 i m, f(u i 1 )=m +n + 4i for 1 i k 1, f(u i )=4i for 1 i k 1, f(v i 1 )=m +n +1 4i for 1 i k 1, f(v i )=4i +for1 i k 1,andf(u 0)=(n + m k). For each vertex label f, the induced edge label f is calculated as follows: f (w i w i+1 )=m i for 1 i m 1, f (u i u i+1 )=n + m 1 i for 1 i k, f (v i v i+1 )=n + m i for 1 i k, f (v 0 u 1 )=n + m 1, f (v 0 v 1 )=n + m, f (u k 1 u 0 )=m +1, f (v k 1 u 0 )=m. Subcase (ii): k>1iseven. Define f : V (C m ) {1,, 3,,p+ q =n +m } as follows: f(w i 1 )=i 1for1 i m, f(w i )=m +1 i for 1 i m, f(u i 1 )=m +n + 4i for 1 i k, f(u i )=4i for 1 i k, f(v i 1 )=m +n 4i for 1 i k, f(v i )=4i +1for1 i k, and f(u 0 )=k. For each vertex label f, the induced edge label f is calculated as follows: f (w i w i+1 )=m i for 1 i m 1, f (u i u i+1 )=n + m 1 i for 1 i k, f (v i v i+1 )=n + m i for 1 i k,

5 Skolem Difference Mean Graphs 47 f (v 0 u 1 )=n + m 1, f (v 0 v 1 )=n + m, f (u k 1 u 0 )=m +1, f (v k 1 u 0 )=m. It can be verified that f is a skolem difference mean labeling. Hence C m is a skolem difference mean graph. Skolem difference mean labelings of C 5 and C 5 are shown in Figure. In the following theorem, we prove that the graph T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i, obtained from the stars K 1,n1,K 1,n,...,K 1,nm by joining the central vertices of K 1,nj and K 1,nj+1 toanewvertexw j for 1 j m 1isanextraskolemdifference mean graph. Theorem.. The graph T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i is an extra skolem difference mean graph. roof. Let u j i (1 i n j) be the pendant vertices and v j (1 j m) be the central vertex of the star K 1,nj (1 j m). Then

6 48 M. Selvi, D. Ramya and. Jeyanthi T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i is a graph obtained by joining v j and v j+1 to a new vertex w j (1 j m 1) by an edge. Define ½ f : V (T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i) 1,, 3,,p+ q = m ¾ n k +4m 3 as follows: f(v j )= m n k +4m 3 (j 1) for 1 j m, f(w 1 )=n 1 +1, f(w j )=n j (n k +1)for j m 1, k= f(u 1 i )=i 1for1 i n 1 f(u j i )=j 1 (n k +1)+i 1for1 i n j and j m. For each vertex label f, the induced edge label f is calculated as follows: Let e j i (1 i n j and 1 j m) be the edges joining the vertices v j with u j i. f (e 1 i )= m n k +m i 1for1 i n 1, f (e j i )=n j +n j+1 + +n m +m j i+1 for 1 i n j and j m, f (v 1 w 1 )=n + n n m +(m 1), f (v j w j )=n j+1 + n j+ + + n m +(m j) for j m 1, f (w 1 v )=n + n n m +m 3, f (w j v j+1 )=n j+1 + n j+ + + n m +(m j) 1for j m 1. It can be verified that T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i is an extra skolem difference mean labeling. Hence T hk 1,n1 : K 1,n : K 1,n3 : : K 1,nm i is an extra skolem difference mean graph. * + Corollary.3. The graph T K 1,n : K 1,n : K 1,n : : K 1,n is an extra skolem difference mean graph. An extra skolem difference mean labeling of T hk 1,7 : K 1,7 : K 1,7 i is showninfigure3.

7 Skolem Difference Mean Graphs 49 The graph T hk 1,n1 K 1,n K 1,nm i is obtained from the stars K 1,n1,K 1,n,,K 1,nm byjoiningaleafofk 1,nj and a leaf of K 1,nj+1 to a new vertex w j (1 j m 1) by an edge. Theorem.4. The graph T hk 1,n1 K 1,n K 1,n3 K 1,nm i is an extra skolem difference mean graph. roof. Let u j i (1 i n j) be the pendant vertices and v j (1 j m) be the central vertex of the star K 1,nj (1 j m). Then T hk 1,n1 K 1,n K 1,n3 K 1,nm i is a graph obtained by joining u j n j and u j+1 1 to a new vertex w j (1 j m 1) by an edge. Define ½ f : V (T hk 1,n1 K 1,n K 1,n3 K 1,nm i) 1,, 3,,p+ q = m ¾ n k +4m 3 as follows: f(v j )= m n k +4m 4j +1for1 j m, f(w j )= m n k +4m 4j 1for1 j m 1, f(u 1 i )=i 1for1 i n 1 f(u j i )=j 1 n k +i 1for1 i n j and j m. Let e j i = v ju j i for 1 i n j and 1 j m. For each vertex label f, the induced edge label f is calculated as follows: f (e 1 i )= m n k +m i 1 for 1 i n 1, f (e j i )=n j + n j n m +m j i +1 for j m 1and

8 50 M. Selvi, D. Ramya and. Jeyanthi 1 i n j, f (u 1 n 1 w 1 )= m n k +m n 1, f (u j n j w j )=n j+1 + n j+ + + n m +(m j) for j m 1, f (w 1 u 1 )=n + n n m +m 3, f (w j u j+1 1 )=n j+1 + n j+ + + n m +m j 1for j m 1. It can be verified that T hk 1,n1 K 1,n K 1,n3 K 1,nm i is an extra skolem difference mean labeling. Hence T hk 1,n1 K 1,n K 1,n3 K 1,nm i is an extra skolem difference mean graph. * + Corollary.5. The graph T K 1,n K 1,n K 1,n K 1,n is a skolem difference mean graph. Askolemdifference mean labeling of T hk 1,6 K 1,6 K 1,6 K 1,6 i is shown in Figure 4.

9 Skolem Difference Mean Graphs 51 Theorem.6. If G is a graph having p vertices and q edges with q>p then G is not a skolem difference mean graph. roof. Let G be a (p, q) graphwithq > p. The minimum possible vertex label of G is 1 and the maximum possible vertex l label m ofl G is mp + q. Therefore, the maximum possible edge label of G is p+q 1 < q 1 = q. That is, G has no edge having the label q and hence G is not a skolem difference mean graph. Corollary.7. ThecompletegraphK n is a skolem difference mean graph if and only if n 3. roof. K is and K 3 is C 3. The graph K n has n vertices and n(n 1) edges. For n 4, n(n 1) >n. By theorem.6, K n, n 4isnotaskolem difference mean graph. Theorem.8. The shrub St(n 1,n,,n m ) is a skolem difference mean graph. roof. Let v 0,v j,u j i (1 j m, 1 i n j ) be the vertices of St(n 1,n,,n m ). Then n E(St(n 1,n,,n m )) = {v 0 v j 1 j m} v j u j o i 1 i n j. ( ) Define f : V (St(n 1,n,,n m )) 1,, 3,,p+ q = m n j +m +1 as follows: f(v 0 )=m + m j=1 n j +1, f(v j )=j 1, 1 j m, f(u j i )=(n j + n j n m )+(j i), 1 i n j 1and1 j m, f(u j n j )=(n j+1 + n j+ + + n m )+j +1for1 j m 1, f(u m n m )=m +1. Let e j i = v ju j i for 1 i n j and 1 j m. For each vertex label f, the induced edge label f is defined as follows: f (v 0 v j )=m + n 1 + n + + n m j +1,1 j m f (e j i )=n j + n j n m i +1,1 i n j and 1 j m. It can be verified that St(n 1,n,,n m )isaskolemdifference mean graph. j=1

10 5 M. Selvi, D. Ramya and. Jeyanthi 5. The skolem difference mean labeling of St(, 5, 5, 3)isshowninFigure Theorem.9. The banana tree Bt(n, n,,n) isaskolemdifference mean graph. roof. Let v 0,v j,u j i (1 j m, 1 i n) be the vertices of Bt(n, n,,n). Then E(Bt(n, n,,n)) = nv 0 u j 1,v ju j o i 1 i n and 1 j m. Define f : V (Bt(n, n,,n)) {1,, 3,,p+ q =m(n +1)+1} as follows: f(v 0 )=m(n +1)+1, f(v j )=m(n 1) + 4j 1, for 1 j m, f(u j 1 )=j 1, 1 j m, f(u j i )=(i )m +j 1, for i n 1and1 j m, f(u j n)=m(n +1) (m j) for1 j m. Lete j i = v ju j i for 1 i n and 1 j m. For each vertex label f, the induced edge label f is calculated as follows: f (v 0 u j 1 )=m(n +1) j +1,for1 j m f (e j i )=m(n i)+j, for1 i n 1and1 j m, f (v j u j n)=m j +1for1 j m.

11 Skolem Difference Mean Graphs 53 It can be verified that Bt(n, n,,n)isaskolemdifference mean graph. Askolemdifference mean labeling of Bt(3,3,3,3,3,3,3,3) is shown in Figure 6. References [1] F. Harary, Graph theory, Addison Wesley, Massachusetts, (197). [] Joseph A. Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics, 17 (014), # DS6. [3] K. Murugan and A. Subramanian, Skolem difference mean labeling of H-graphs, International Journal of Mathematics and Soft Computing, Vol.1, No. 1, pp , (011).

12 54 M. Selvi, D. Ramya and. Jeyanthi [4] K. Murugan and A. Subramanian, Labeling of subdivided graphs, American Journal of Mathematics and Sciences, Vol.1, No. 1, pp , (01). [5] K. Murugan and A. Subramanian, Labeling of finite union of paths, International Journal of Mathematics and Soft Computing, Vol.,No. 1, pp , (01). [6] D. Ramya, M. Selvi and R. Kalaiyarasi, On skolem difference mean labeling of graphs, International Journal of Mathematical Archive, 4(1), pp , (013). [7] D. Ramya and M. Selvi, On skolem difference mean labeling of some trees, International Journal of Mathematics and Soft Computing, Vol.4, No., pp , (014). [8] S. Somasundaram and R. onraj, Mean labelings of graphs, National Academy Science Letter, 6, pp , (003). M. Selvi Department of Mathematics Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-68 15, Tamilnadu, India D. Ramya Department of Mathematics Dr. Sivanthi Aditanar College of Engineering, Tiruchendur-68 15, Tamilnadu, India aymar padma@yahoo.co.in and. Jeyanthi Research Centre, Department of Mathematics Govindammal Aditanar College for Women Tiruchendur-68 15, Tamilnadu, India jeyajeyanthi@rediffmail.com