Super Mean Labeling of Some Classes of Graphs

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1 International J.Math. Combin. Vol.1(01), Super Mean Labeling of Some Classes of Graphs P.Jeyanthi Department of Mathematics, Govindammal Aditanar College for Women Tiruchendur-68 15, Tamil Nadu, India D.Ramya Department of Mathematics, Dr. Sivanthi Aditanar College of Engineering Tiruchendur , Tamil Nadu, India aymar Abstract: Let G be a (p,q) graph and f : V (G) {1,,3,..., p + q} be an injection. For each edge e = uv, let f (e) = (f(u) + f(v))/ if f(u) + f(v) is even and f (e) = (f(u) + f(v)+1)/ if f(u)+f(v) is odd. Then f is called a super mean labeling if f(v ) {f (e) : e E(G)} = {1,, 3,..., p+q}. A graph that admits a super mean labeling is called a super mean graph. In this paper we prove that S(P n K 1), S(P P 4), S(B n,n), B n,n : P m, C n K, n 3, generalized antiprism A m n and the double triangular snake D(T n) are super mean graphs. Key Words: Smarandachely super m-mean labeling, Smarandachely super m-mean graph, super mean labeling, super mean graph. AMS(010): 05C78 1. Introduction 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. The disjoint union of two graphs G 1 and G is the graph G 1 G with V (G 1 G ) = V (G 1 ) V (G ) and E(G 1 G ) = E(G 1 ) E(G ). The disjoint union of m copies of the graph G is denoted by mg. The corona G 1 G of the graphs G 1 and G is obtained by taking one copy of G 1 (with p vertices) and p copies of G and then joining the i th vertex of G 1 to every vertex in the i th copy of G. Armed crown C n ΘP m is a graph obtained from a cycle C n by identifying the pendent vertex of a path P m at each vertex of the cycle. Bi-armed crown is a graph obtained from a cycle C n by identifying the pendant vertices of two vertex disjoint paths of equal length m 1 at each vertex of the cycle. We denote a bi-armed crown by C n ΘP m, where P m is a path of length m 1. The double triangular snake D(T n ) is the graph obtained from the path v 1, v, v 3,..., v n by joining v i and v i+1 with two new vertices i i and w i for 1 i n 1. The bistar B m,n is a graph obtained from 1 Received October 19, 011. Accepted March 10, 01.

2 84 P.Jeyanthi and D.Ramya K by joining m pendant edges to one end of K and n pendant edges to the other end of K. The generalized prism graph C n P m has the vertex set V = {v j i : 1 i n and 1 j m} and the edge set E = {v j i vj i+1, vj n vj 1 : 1 i n 1 and 1 j m} {vj i vj+1 i 1, vj 1 vj+1 n : i n and 1 j m 1}. The generalized antiprism A m n is obtained by completing the generalized prism C n P m by adding the edges v j i vj+1 i for 1 i n and 1 j m 1. Terms and notations not defined here are used in the sense of Harary [1].. Preliminary Results Let G be a graph and f : V (G) {1,, 3,, V + E(G) } be an injection. For each edge e = uv and an integer m, the induced Smarandachely edge m-labeling fs is defined by f(u) + f(v) fs (e) =. m Then f is called a Smarandachely super m-mean labeling if f(v (G)) {f (e) : e E(G)} = {1,, 3,, V + E(G) }. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph. Particularly, if m =, we know that f (e) = f(u)+f(v) if f(u) + f(v) is even; f(u)+f(v)+1 if f(u) + f(v) is odd. Such a labeling f is called a super mean labeling of G if f(v (G)) (f (e) : e E(G)} = {1,, 3,..., p + q}. A graph that admits a super mean labeling is called a super mean graph. The concept of super mean labeling was introduced in [7] and further discussed in [-6]. We use the following results in the subsequent theorems. Theorem.1([7]) The bistar B m,n is a super mean graph for m = n or n + 1. Theorem.([]) The graph B n,n : w, obtained by the subdivision of the central edge of B n,n with a vertex w, is a super mean graph. Theorem.3([]) The bi-armed crown C n ΘP m is a super mean graph for odd n 3 and m. Theorem.4([7]) Let G 1 = (p 1, q 1 ) and G = (p, q ) be two super mean graphs with super mean labeling f and g respectively. Let f(u) = p 1 +q 1 and g(v) = 1. Then the graph (G 1 ) f (G ) g obtained from G 1 and G by identifying the vertices u and v is also a super mean graph. 3. Super Mean Graphs If G is a graph, then S(G) is a graph obtained by subdividing each edge of G by a vertex. Theorem 3.1 The graph S(P n K 1 ) is a super mean graph.

3 Super Mean Labeling of Some Classes of Graphs 85 Proof Let V (P n K 1 ) = {u i, v i : 1 i n}. Let x i (1 i n) be the vertex which divides the edge u i v i (1 i n) and y i (1 i n 1) be the vertex which divides the edge u i u i+1 (1 i n 1). Then V (S(P n K 1 )) = {u i, v i, x i, y j : 1 i n, 1 j n 1}. Define f : V (S(P n K 1 )) {1,, 3,..., p + q = 8n 3} by f(v 1 ) = 1; f(v ) = 14; f(v +i ) = i for 1 i n 4; f(v n 1 ) = 8n 11; f(v n ) = 8n 10; f(x 1 ) = 3; f(x 1+i ) = 3 + 8i for 1 i n ; f(x n ) = 8n 7; f(u 1 ) = 5; f(u ) = 9; f(u +i ) = 9 + 8i for 1 i n 3; f(u n ) = 8n 5; f(y i ) = 8i 1 for 1 i n ; f(y n 1 ) = 8n 3. It can be verified that f is a super mean labeling of S(P n K 1 ). Hence S(P n K 1 ) is a super mean graph. Example 3. The super mean labeling of S(P 5 K 1 ) is given in Fig.1. Fig.1 Theorem 3. The graph S(P P n ) is a super mean graph. Proof Let V (P P n ) = {u i, v i : 1 i n}. Let u 1 i, v1 i (1 i n 1) be the vertices which divide the edges u i u i+1, v i v i+1 (1 i n 1) respectively. Let w i (1 i n) be the vertex which divides the edge u i v i. That is V (S(P P n )) = {u i, v i, w i : 1 i n} {u 1 i, v1 i : 1 i n 1}. Define f : V (S(P P n )) {1,, 3,..., p + q = 11n 6} by f(u 1 ) = 1; f(u ) = 9; f(u 3 ) = 7; f(u i ) = f(u i 1 ) + 5 for 4 i n and i is even f(u i ) = f(u i 1 ) + 17 for 4 i n and i is odd f(v 1 ) = 7; f(v ) = 16; f(v i ) = f(v i 1 ) + 5 for 3 i n and i is odd f(v i ) = f(v i 1 ) + 17 for 3 i n and i is even f(w 1 ) = 3; f(w ) = 1; f(w +i ) = i for 1 i n ; f(u 1 1 ) = 6; f(u1 ) = 4;

4 86 P.Jeyanthi and D.Ramya f(u 1 i ) = f(u1 i 1 ) + 6 for 3 i n 1 and i isodd f(u 1 i ) = f(u1 i 1 ) + 16 for 3 i n 1 and i iseven f(v 1 1) = 13; f(v 1 i ) = f(v 1 i 1) + 6 for i n 1 and i iseven f(vi 1 ) = f(v1 i 1 ) + 16 for i n 1 and i isodd. It is easy to check that f is a super mean labeling of S(P P n ). Hence S(P P n ) is a super mean graph. Example 3.4 The super mean labeling of S(P P 6 ) is given in Fig.. Fig. Theorem 3.5 The graph S(B n,n ) is a super mean graph. Proof Let V (B n,n ) = {u, u i, v, v i : 1 i n} and E(B n,n ) = {uu i, vv i, uv : 1 i n}. Let w, x i, y i, (1 i n) be the vertices which divide the edges uv, uu i, vv i (1 i n) respectively. Then V (S(B n,n )) = {u, u i, v, v i, x i, y i, w : 1 i n} and E(S(B n,n )) = {ux i, x i u i, uw, wv, vy i, y i v i : 1 i n}. Define f : V (S(B n,n )) {1,, 3,..., p + q = 8n + 5} by f(u) = 1; f(x i ) = 8i 5 for 1 i n; f(u i ) = 8i 3 for 1 i n; f(w) = 8n + 3; f(v) = 8n + 5; f(y i ) = 8i 1 for 1 i n; f(v i ) = 8i + 1 for 1 i n. It can be verified that f is a super mean labeling of S(B n,n ). Hence S(B n,n ) is a super mean graph. Example 3.6 The super mean labeling of S(B n,n ) is given in Fig.3. Fig.3

5 Super Mean Labeling of Some Classes of Graphs 87 Next we prove that the graph B n,n : P m is a super mean graph. B m,n : P k is a graph obtained by joining the central vertices of the stars K 1,m and K 1,n by a path P k of length k 1. Theorem 3.7 The graph B n,n : P m is a super mean graph for all n 1 and m > 1. Proof Let V ( B n,n : P m ) = {u i, v i, u, v, w j : 1 i n, 1 j m with u = w 1, v = w m } and E( B n,n : P m ) = {uu i, vv i, w j w j+1 : 1 i n, 1 j m 1}. Case 1 n is even. Subcase 1 m is odd. By Theorem., B n,n : P 3 is a super mean graph. For m > 3, define f : V ( B n,n : P m ) {1,, 3,..., p + q = 4n + m 1} by f(u) = 1; f(u i ) = 4i 1 for 1 i n and for i n + 1; f(u n +1 ) = n + ; f(v i ) = 4i + 1 for 1 i n; f(v) = 4n + 3; f(w ) = 4n + 4; f(w 3 ) = 4n + 9; f(w 3+i ) = 4n i for 1 i m 5 ; f(w m+3) = 4n + m 4 m 5 f(w m+3 +i ) = 4n + m 4 4i for 1 i. It can be verified that f is a super mean labeling of B n,n : P m. Subcase m is even. By Theorem.1, B n,n : P is a super mean graph. For m >, define f : V ( B n,n : P m ) {1,, 3,..., p + q = 4n + m 1} by f(u) = 1; f(u i ) = 4i 1 for 1 i n and for i n + 1; f(u n +1 ) = n + ; f(v i ) = 4i + 1 for 1 i n; f(v) = 4n + 3; f(w ) = 4n + 4; f(w +i ) = 4n i for 1 i m 4 ; f(w m+ ) = 4n + m + 3; m 4 f(w m+ +i ) = 4n + m i for 1 i. It can be verified that f is a super mean labeling of B n,n : P m. Case n is odd. Subcase 1 m is odd. By Theorem.1, B n,n : P is a super mean graph. For m >, define f : V ( B n,n : P m )

6 88 P.Jeyanthi and D.Ramya {1,, 3,..., p + q = 4n + m 1} by f(u) = 1; f(v) = 4n + 3; f(u i ) = 4i 1 for 1 i n; f(v i ) = 4i + 1 for 1 i n and for i n + 1 ; f(v n+1) = n + ; f(w ) = 4n + 4; f(w +i ) = 4n i for 1 i m 4 ; f(w m+) = 4n + m + 3; m 4 f(w m+ +i ) = 4n + m i for 1 i. It can be verified that f is a super mean labeling of B n,n : P m. Subcase m is even. By Theorem., B n,n : P 3 is a super mean graph. For m > 3, define f : V ( B n,n : P m ) {1,, 3,..., p + q = 4n + m 1} by f(u) = 1; f(v) = 4n + 3; f(u i ) = 4i 1 for 1 i n; f(v i ) = 4i + 1 for 1 i n and for i n + 1 f(w ) = 4n + 4; f(w 3 ) = 4n + 9; ; f(v n+1) = n + ; f(w 3+i ) = 4n i for 1 i m 5 ; f(w m+3 ) = 4n + m 4; m 5 f(w m+3 +i ) = 4n + m 4 i for 1 i. It can be verified that f is a super mean labeling of B n,n : P m. Hence B n,n : P m is a super mean graph for all n 1 and m > 1. Example 3.8 The super mean labeling of B 4,4 : P 5 is given in Fig.4. Fig.4 Theorem 3.9 The corona graph C n K is a super mean graph for all n 3. Proof Let V (C n ) = {u 1, u,...,u n } and V (C n K ) = {u i, v i, w i : 1 i n}. Then E(C n K ) = {u i u i+1, u n u 1, u j v j, u j w j : 1 i n 1 and 1 j n}. Case 1 n is odd.

7 Super Mean Labeling of Some Classes of Graphs 89 The proof follows from Theorem.3 by taking m =. Case n is even. Take n = k for some k. Define f : V (C n K ) {1,, 3,..., p + q = 6n} by f(u i ) = 6i 3 for 1 i k 1; f(u k ) = 6k ; f(u k+i ) = 6k + 6i for 1 i k ; f(u k 1 ) = 1k ; f(u k ) = 1k 9; f(v i ) = 6i 5 for 1 i k 1; f(v k ) = 6k 6; f(v k+1 ) = 6k + ; f(v k+1+i ) = 6k + + 6i for 1 i k 3; f(v k 1 ) = 1k; f(v k ) = 1k 6; f(w i ) = 6i 1 for 1 i k 1; f(w k ) = 6k; f(w k+i ) = 6k + 6i for 1 i k ; f(w k 1 ) = 1k 4; f(w k ) = 1k 11. It can be verified that f(v ) (f (e) : e E} = {1,, 3,..., 6n}. Hence C n K is a super mean graph. Example 3.10 The super mean labeling of C 8 K is given in Fig.5. Fig.5 Theorem 3.11 The double triangular snake D(T n ) is a super mean graph. Proof We prove this result by induction on n. A super mean labeling of G 1 = D(T ) is given in Fig.6. Fig.6

8 90 P.Jeyanthi and D.Ramya Therefore the result is true for n =. Let f be the super mean labeling of G 1 as in the above figure. Now D(T 3 ) = (G 1 ) f (G 1 ) f, by Theorem.4, D(T 3 ) is a super mean graph. Therefore the result is true for n = 3. Assume that D(T n 1 ) is a super mean graph with the super mean labeling g. Now by Theorem.4, (D(T n 1 )) g (G 1 ) f = D(T n ) is a super mean graph. Therefore the result is true for n. Hence by induction principle the result is true for all n. Thus D(T n ) is a super mean graph. Example 3.1 The super mean labeling of D(T 6 ) is given in Fig.7. Fig.7 Theorem 3.13 The generalized antiprism A m n except for n = 4. is a super mean graph for all m, n 3 Proof Let V (A m n ) = {vj i : 1 i n, 1 j m} and E(Am n ) = {vj i vj i+1, vj n vj 1 : 1 i n 1, 1 j m} {v j i vj+1 i 1, vj 1 vj+1 n : i n, 1 j m 1} {v j i vj+1 i : 1 i n and 1 j m 1}. Case 1 n is odd. Define f : V (A m n ) {1,, 3,..., p + q = 4mn n} by f(v j n + 1 i ) = 4(j 1)n + i 1 for 1 i and 1 j m; f(v j n+3) = 4(j 1)n + n + 3 for 1 j m; f(v j n 3 n+3 +i) = 4(j 1)n + n i for 1 i and 1 j m. Then f is a super mean labeling of A m n. Hence A m n is a super mean graph. Case n is even and n 4. Define f : V (A m n ) {1,, 3,..., p + q = 4mn n} by f(v j 1 ) = 4(j 1)n + 1 for 1 j m; f(vj ) = 4(j 1)n + 3 for 1 j m; f(v j 3 ) = 4(j 1)n + 7 for 1 j m; f(vj 4 ) = 4(j 1)n + 1 for 1 j m; n 6 ) = 4(j 1)n i for 1 j m and 1 i ; f(v j 4+i f(v j n+ n 6 +i) = 4(j 1)n + n + 1 4i for 1 j m and 1 i ; f(v j n 1 ) = 4(j 1)n + 9 for 1 j m; f(vj n ) = 4(j 1)n + 6 for 1 j m.

9 Super Mean Labeling of Some Classes of Graphs 91 Then f is a super mean labeling of A m n. Hence Am n is a super mean graph. Example 3.14 The super mean labeling of A 3 6 is given in Fig.8. Fig.8 References [1] F.Harary, Graph theory, Addison Wesley, Massachusetts, (197). [] P.Jeyanthi, D.Ramya and P.Thangavelu, On super mean labeling of graphs, AKCE Int. J. Graphs. Combin., 6(1) (009), [3] P.Jeyanthi, D.Ramya and P.Thangavelu, Some constructions of k-super mean graphs, International Journal of Pure and Applied Mathematics, 56(1), [4] P.Jeyanthi, D.Ramya and P.Thangavelu, On super mean labeling of some graphs, SUT Journal of Mathematics, 46(1) (010), [5] P.Jeyanthi and D.Ramya, Super mean graphs, Utilitas Math., (To appear). [6] R.Ponraj and D.Ramya, On super mean graphs of order 5, Bulletin of Pure and Applied Sciences, 5(1) (006), [7] D.Ramya, R.Ponraj and P.Jeyanthi, Super mean labeling of graphs, Ars Combin., (To appear).