Some Results on Super Mean Graphs

International J.Math. Combin. Vol.3 (009), 8-96 Some Results on Super Mean Graphs R. Vasuki 1 and A. Nagarajan 1 Department of Mathematics, Dr.Sivanthi Aditanar College of Engineering, Tiruchendur - 68 15, Tamil Nadu, India. Department of Mathematics, V.O. Chidambaram College, Thoothukudi - 68 00, Tamil Nadu, India. E-mail: vasukisehar@yahoo.co.in, nagarajan.voc@gmail.com Abstract: Let G be a graph and f : V (G) {1,, 3,..., p + q} be an injection. For each edge e = uv and an integer m, the induced Smarandachely edge m-labeling f S is defined by fs(e) f(u) + f(v) =. m Then f is called a Smarandachely super m-mean labeling if f(v (G)) {f (e) : e E(G)} = {1,,3,..., p + q}. Particularly, in the case of 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 is usually called a super mean labeling. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph. In this paper, we prove that the H-graph, corona of a H-graph, G S where G is a H-graph, the cycle C n for n 3, corona of the cycle C n for n 3, mc n-snake for m 1, n 3 and n 4, the dragon P n(c m) for m 3 and m 4 and C m P n for m = 3, 5 are super mean graphs, i.e., Smarandachely super -mean graphs. Keywords: Labeling, Smarandachely super mean labeling, Smarandachely super m-mean graph, super mean labeling, super mean graphs AMS(000): 05C78 1. Introduction Throughout this paper, by a graph we mean a finite, undirected, simple graph. Let G(V, E) be a graph with p vertices and q edges. For notations and terminology we follow [1]. Let G 1 and G be any two graphs with p 1 and p vertices respectively. Then the Cartesian 1 Received July 4, 009. Accepted Aug.8, 009.

Some Results on Super Mean Graphs 83 product G 1 G has p 1 p vertices which are {(u, v)/u G 1, v G }. The edges are obtained as follows: (u 1, v 1 ) and (u, v ) are adjacent in G 1 G if either u 1 = u and v 1 and v are adjacent in G or u 1 and u are adjacent in G 1 and v 1 = v. The corona of a graph G on p vertices v 1, v,..., v p is the graph obtained from G by adding p new vertices u 1, u,..., u p and the new edges u i v i for 1 i p, denoted by G K 1. For a graph G, the -corona of G is the graph obtained from G by identifying the center vertex of the star S at each vertex of G, denoted by G S. The baloon of a graph G, P n (G) is the graph obtained from G by identifying an end vertex of P n at a vertex of G. P n (C m ) is called a dragon. The join of two graphs G and H is the graph obtained from G H by joining each vertex of G with each vertex of H by means of an edge and it is denoted by G + H. A path of n vertices is denoted by P n and a cycle on n vertices is denoted by C n. K 1,m is called a star, denoted by S m. The bistar B m,n is the graph obtained from K by identifying the center vertices of K 1,m and K 1,n at the end vertices of K respectively, denoted by B(m). A triangular snake T n is obtained from a path v 1 v... v n by joining v i and v i+1 to a new vertex w i for 1 i n 1, that is, every edge of a path is replaced by a triangle C 3. We define the H-graph of a path P n to be the graph obtained from two copies of P n with vertices v 1, v,..., v n and u 1, u,..., u n by joining the vertices v n+1 and u n+1 if n is odd and the vertices v n +1 and u n if n is even and a cyclic snake mc n the graph obtained from m copies of C n by identifying the vertex v (k+)j in the j th copy at a vertex v 1j+1 in the (j + 1) th copy if n = k + 1 and identifying the vertex v (k+1)j in the j th copy at a vertex v 1j+1 in the (j + 1) th copy if n = k. A vertex labeling of G is an assignment f : V (G) {1,, 3,..., p + q} be an injection. For a vertex labeling f, the induced Smarandachely edge m-labeling fs for an edge e = uv, an integer m 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,..., p + q}. Particularly, in the case of 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 is usually called a super mean labeling. A graph that admits a Smarandachely super mean m-labeling is called Smarandachely super m-mean graph, particularly, super mean graph if m =. A super mean labeling of the graph P6 is shown in Fig.1.1. Fig.1.1

84 R. Vasuki and A. Nagarajan The concept of mean labeling was first introduced by S. Somasundaram and R. Ponraj [7]. They have studied in [4,5,7,8] the mean labeling of some standard graphs. The concept of super mean labeling was first introduced by R. Ponraj and D. Ramya []. They have studied in [,3] the super mean labeling of some standard graphs like P n, C n+1, n 1, K n (n 3), K 1,n (n 3), T n, C m P n (m 3, n 1), B m,n (m = n, n + 1) etc. They have proved that the union of two super mean graph is super mean graph and C 4 is not a super mean graph. Also they determined all super mean graph of order 5. In this paper, we establish the super meanness of the graph C n for n 3, the H-graph, Corona of a H- graph, -corona of a H-graph, corona of cycle C n for n 3, mc n -snake for m 1, n 3 and n 4, the dragon P n (C m ) for m 3 and m 4 and C m P n for m = 3, 5.. Results Theorem.1 The H-graph G is a super mean graph. Proof Let v 1, v,..., v n and u 1, u,..., u n be the vertices of the graph G. We define a labeling f : V (G) {1,,..., p + q} as follows: f(v i ) = i 1, 1 i n f(u i ) = n + i 1, 1 i n For the vertex labeling f, the induced edge labeling f is defined as follows: f (v i v i+1 ) = i, 1 i n 1 f (u i u i+1 ) = n + i, 1 i n 1 f (v n+1u n+1) = n if n is odd f (v n +1 u n) = n if n is even Then clearly it can be verified that the H-graph G is a super mean graph. For example the super mean labelings of H-graphs G 1 and G are shown in Fig..1. Fig..1

Some Results on Super Mean Graphs 85 Theorem. If a H-graph G is a super mean graph, then G K 1 is a super mean graph. Fig.. Proof Let f be a super mean labeling of G with vertices v 1, v,..., v n and u 1, u,..., u n. Let v 1, v,...,v n and u 1, u,..., u n be the corresponding new vertices in G K 1. We define a labeling g : V (G K 1 ) {1,,..., p + q} as follows: g(v i ) = f(v i ) + i, 1 i n g(u i ) = f(u i ) + n + i, 1 i n g(v 1 ) = f(v 1) g(v i ) = f(v i) + i 3, i n g(u i ) = f(u i) + n + i 3, 1 i n For the vertex labeling g, the induced edge labeling g is defined as follows:

86 R. Vasuki and A. Nagarajan g (v i v i+1 ) = f (v i v i+1 ) + i + 1, 1 i n 1 g (u i u i+1 ) = f (u i u i+1 ) + n + i + 1, 1 i n 1 g (v i v i ) = f(v i) + i 1, 1 i n g (u i u i ) = f(u i) + n + i 1, 1 i n g (v n+1 u n+1 ) = f ((v n+1 g (v n +1 u n ) = f ((v n u n+1) + 1 if n is odd ) + 1 if n is even +1 u n It can be easily verified that g is a super mean labeling and hence G K 1 is a super mean graph. For example the super mean labeling of H-graphs G 1, G, G 1 K 1 and G K 1 are shown in Fig... Theorem.3 If a H-graph G is a super mean graph, then G S is a super mean graph. Proof Let f be a super mean labeling of G with vertices v 1, v,..., v n and u 1, u,..., u n. Let v 1, v,...,v n, v 1,v,..., v n, u 1, u,...,u n and u 1, u,...,u n be the corresponding new vertices in G S. We define g : V (G S ) {1,,..., p + q} as follows: g(v i ) = f(v i ) + 4i, 1 i n g(v i ) = f(v i) + 4i 4, 1 i n g(v i ) = f(v i) + 4i, 1 i n g(u i ) = f(u i ) + 4n + 4i, 1 i n g(u i ) = f(u i) + 4n + 4i 4, 1 i n g(u i ) = f(u i) + 4n + 4i, 1 i n For the vertex labeling g, the induced edge labeling g is defined as follows: g (v n+1 u n+1 ) = 3f (v n+1 g (v n +1 u n ) = 3f (v n u n+1) if n is odd ) if n is even +1 u n g (v i v i+1 ) = f (v i v i+1 ) + 4i, 1 i n 1 g (v i v i ) = f(v i) + 4i 3, 1 i n g (v i v i ) = f(v i) + 4i 1, 1 i n g (u i u i+1 ) = f (u i u i+1 ) + 4n + 4i 1 i n 1 g (u i u i ) = f(u i) + 4n + 4i 3, 1 i n g (u i u i ) = f(u i) + 4n + 4i 1, 1 i n It can be easily verified that g is a super mean labeling and hence G S is a super mean graph. For example the super mean labelings of G 1 S and G S are shown in Fig..3.

Some Results on Super Mean Graphs 87 Fig..3 Theorem.4 Cycle C n is a super mean graph for n 3. Proof Let C n be a cycle with vertices u 1, u,..., u n and edges e 1, e,..., e n. Define f : V (C n ) {1,,..., p + q} as follows: f(u 1 ) = 1 f(u i ) = 4i 5, i n f(u n+j ) = 4n 3j + 3, 1 j f(u n+j+ ) = 4n 4j, 1 j n For the vertex labeling f, the induced edge labeling f is defined as follows:

88 R. Vasuki and A. Nagarajan f (e 1 ) = f (e i ) = 4i 3, i n 1 f (e n ) = 4n, f (e n+1 ) = 4n 1, f (e n+j+1 ) = 4n 4j, 1 j n 1 It can be easily verified that f is a super mean labeling and hence C n is a super mean graph. For example the super mean labeling of C 10 is shown in Fig..4. Fig..4 Remark.5 In [], it was proved that C n+1, n 1 is a super mean graph and C 4 is not a super mean graph and hence the cycle C n is a super mean graph for n 3 and n 4. Theorem.6 Corona of a cycle C n is a super mean graph for n 3. Proof Let C n be a cycle with vertices u 1, u,..., u n and edges e 1, e,..., e n. Let v 1, v,..., v n be the corresponding new vertices in C n K 1 and E i be the edges joining u i v i, i = 1 to n. Define f : V (C n K 1 ) {1,,..., p + q} as follows: Case i When n is odd, n = m + 1, m = 1,, 3,... f(u 1 ) = 3 f(u i ) = f(v 1 ) = 1 f(v i ) = 5 + 8(i ) i m + 1 1 + 8(m + 1 i) m + i m + 1 7 + 8(i ) i m + 1 10 + 8(m + 1 i) m + i m + 1 For the vertex labeling f, the induced edge labeling f is defined as follows: f (e 1 ) = 4

Some Results on Super Mean Graphs 89 f 9 + 8(i ) i m + 1 (e i ) = 8 + 8(m + 1 i) m + i m + 1 f (E 1 ) = f 6 + 8(i ) i m + 1 (E i ) = 11 + 8(m + 1 i) m + i m + 1 Case ii When n is even, n = m, m =, 3,... f(u 1 ) = 3 f(u i ) = 5 + 8(i ), i m f(u m+1 ) = 8m, f(u i ) = 1 + 8(m i), m + i m f(v 1 ) = 1 f(v i ) = 7 + 8(i ), i m f(v m+1 ) = 8m, f(v m+ ) = 8m 7, f(v i ) = 10 + 8(m i), m + 3 i m For the vertex labeling f, the induced edge labeling f is defined as follows: f (e 1 ) = 4 f (e i ) = 9 + 8(i ), i m 1 f (e m ) = 8m 6, f (e m+1 ) = 8m 3, f (e i ) = 8 + 8(m i), m + i m f (E 1 ) = f (E i ) = 6 + 8(i ), i m f (E m+1 ) = 8m 1 f (E i ) = 11 + 8(m i), m + i m It can be easily verified that f is a super mean labeling and hence C n K 1 is a super mean graph. For example the super mean labelings of C 7 K 1 and C 8 K 1 are shown in Fig..5.

90 R. Vasuki and A. Nagarajan Fig..5 Remark.7 C 4 is not a super mean graph, but C 4 K 1 is a super mean graph. Theorem.8 The graph mc n - snake, m 1, n 3 and n 4 has a super mean labeling. Proof We prove this result by induction on m. Let v 1j, v j,..., v nj be the vertices and e 1j, e j,...,e nj be the edges of mc n for 1 j m. Let f be a super mean labeling of the cycle C n.

Some Results on Super Mean Graphs 91 When m = 1, by Remark 1.5, C n is a super mean graph, n 3, n 4. Hence the result is true when m = 1. Let m =. The cyclic snake C n is the graph obtained from copies of C n by identifying the vertex v (k+)1 in the first copy of C n at a vertex v 1 in the second copy of C n when n = k+1 and identifying the vertex v (k+1)1 in the first copy of C n at a vertex v 1 in the second copy of C n when n = k. Fig..6 Define a super mean labeling g of C n as follows: For 1 i n, g(v i1 ) = f(v i1 ) g(v i ) = f(v i1 ) + n 1 g (e i1 ) = f (e i1 ) g (e i ) = f (e i1 ) + n 1. Thus, C n -snake is a super mean graph. Assume that mc n -snake is a super mean graph for any m 1. We will prove that (m+1)c n - snake is a super mean graph. Super mean labeling g of (m + 1)C n is defined as follows: g(v ij ) = f(v i1 ) + (j 1)(n 1), 1 i n, j m g(v im+1 ) = f(v i1 ) + m(n 1), 1 i n

9 R. Vasuki and A. Nagarajan For the vertex labeling g, the induced edge labeling g is defined as follows: g (e ij ) = f (e i1 ) + (j 1)(n 1), 1 i n, j m g (e im+1 ) = f (e i1 ) + m(n 1), 1 i n Then it is easy to check the resultant labeling g is a super mean labeling of (m + 1)C n -snake. For example the super mean labelings of 4C 6 -snake and 4C 5 - snake are shown in Fig..6. Theorem.9 If G is a super mean graph then P n (G) is also a super mean graph. Proof Let f be a super mean labeling of G. Let v 1, v,..., v p be the vertices and e 1, e,...,e q be the edges of G and let u 1, u,..., u n and E 1, E,...,E n 1 be the vertices and edge of P n respectively. We define g on P n (G) as follows: g(v i ) = f(v i ), 1 i p. g(u j ) = p + q + j, 1 j n. For the vertex labeling g, the induced edge labeling g is defined as follows: g (e i ) = f(e i ) 1 i p. g (E j ) = p + q + j 1, 1 j n 1. Then g is a super mean labeling of P n (G). Corollary 1.10 Dragon P n (C m ) is a super mean graph for m 3 and m 4. Proof Since C m is a super mean graph for m 3 and m 4, by using the above theorem, P n (C m ) for m 3 and m 4 is also a super mean graph. For example, the super mean labeling of P 5 (C 6 ) is shown in Fig..7. Fig..7 Remark.11 The converse of the above theorem need not be true. For example consider the graph C 4. P n (C 4 ) for n 3 is a super mean graph but C 4 is not a super mean graph. The super mean labeling of the graph P 4 (C 4 ) is shown in Fig..8

Some Results on Super Mean Graphs 93 Fig..8 Theorem.1 C m P n for n 1, m = 3, 5 are super mean graphs. Proof Let V (C m P n ) = {v ij : 1 i m, 1 j n} and E(C m P n ) = {e ij : e ij = v ij v (i+1)j, 1 j n, 1 i m} {E ij : E ij = v ij v ij+1, 1 j n 1, 1 i m} where i + 1 is taken modulo m. Case i m = 3 First we label the vertices of C 1 3 and C 3 as follows: f(v 11 ) = 1 f(v i1 ) = 3i 3, i 3 f(v i ) = 1 + 3(i 1), 1 i f(v 3 ) = 10 For the vertex labeling f, the induced edge labeling f is defined as follows: f (e i1 ) = + 3(i 1), 1 i f (e 31 ) = 4 f (e 1 ) = 14 f (e i ) = 13 (i ), i 3 f (E i1 ) = 7 + (i 1), 1 i f (E 31 ) = 8

94 R. Vasuki and A. Nagarajan Fig..9 If the vertices and edges of C j 1 3 and C j 3 are labeled then the vertices and edges of Cj+1 3 and C j+ 3 are labeled as follows:

Some Results on Super Mean Graphs 95 f(v ij+1 ) = f(v ij 1 ) + 18, 1 i 3, 1 j n 1 if n is odd and 1 j n if n is even f(v ij+ ) = f(v ij ) + 18, 1 i 3, 1 j n 3 if n is odd and 1 j n if n is even. f (e ij+1 ) = f (e ij 1 ) + 18, 1 i 3, 1 j n 1 if n is odd and 1 j n if n is even f (e jj+ ) = f (e ij ) + 18, 1 i 3, 1 j n 3 if n is odd and f (E ij+1 ) = f (E ij 1 ) + 18, 1 j n if n is even 1 i 3, 1 j n 3 if n is odd and 1 j n if n is even f (E ij+ ) = f (E ij ) + 18, 1 i 3, 1 j n 3 if n is odd and 1 j n 4 if n is even Case ii m = 5. First we Label the vertices of C 1 5 and C 5 as follows: f(v 11 ) = 1 4i 5, i 3 f(v i1 ) = 10 4(i 4), 4 i 5 f(v 1 ) = 1 5 3(i ), i 3 f(v i ) = 16 + (i 4) 4 i 5 For the vertex labeling f, the induced edge labeling f is defined as follows: f (e i1 ) = + 3(i 1), 1 i f (e 31 ) = 9 f (e i1 ) = 8 4(i 4), 4 i 5 3 + (i 1), 1 i f (e i ) = 19 (i 3), 3 i 4 f (e 5 ) = 0, f (E 11 ) = 11 14 + (i ), i 3 f (E i1 ) = 13 (i 4), 4 i 5 If the vertices and edges of C j 1 5 and C j 5 are labeled then the vertices and edges of Cj+1 5 and C j+ 5 are labeled as follows:

96 R. Vasuki and A. Nagarajan f(v ij+1 ) = l(v ij 1 ) + 30, 1 i 5, 1 j n if n is even and 1 j n 1 if n is odd f(v ij+ ) = l(v ij ) + 30, 1 i 5, 1 j n if n is even and 1 j n 3 if n is odd. f (E ij+1 ) = f (E ij 1 ) + 30, 1 i 5, 1 j n 3 if n is odd and f (E ij+ ) = f (E ij ) + 30, 1 i 5, 1 j n if n is even 1 j n 3 if n is odd and 1 j n 4 if n is even f (e ij+1 ) = f (e ij 1 ) + 30, 1 i 5, 1 j n if n is even and f (e ij+ ) = f (e ij ) + 30, 1 i 5, 1 j n 1 if n is odd 1 j n if n is even and 1 j n 3 if n is odd. Then it is easy to check that the labeling f is a super mean labeling of C 3 P n and C 5 P n. For example the super mean labeling of C 3 P 5 and C 5 P 4 are shown in Fig..9. 3. Open Problems We present the following open problem for further research. Open Problem. For what values of m (except 3,5) the graph C m P n is super mean graph. References [1] R. Balakrishnan and N. Renganathan, A Text Book on Graph Theory, Springer Verlag, 000. [] R. Ponraj and D. Ramya, Super Mean labeling of graphs, Preprint. [3] R. Ponraj and D. Ramya, On super mean graphs of order 5, Bulletin of Pure and Applied Sciences, (Section E Maths and Statistics) 5E(006), 143-148. [4] R. Ponraj and S. Somasundaram, Further results on mean graphs, Proceedings of Sacoeference, August 005. 443-448. [5] R. Ponraj and S. Somasundaram, Mean graphs obtained from mean graphs, Journal of Discrete Mathematical Sciences & Cryptography, 11()(008), 39-5. [6] Selvam Avadayappan and R. Vasuki, Some results on mean graphs, Ultra Scientist of Physical Sciences, 1(1)M (009), 73-84. [7] S. Somasundaram and R. Ponraj, Mean labelings of graphs, National Academy Science letter, 6 (003), 10-13. [8] S. Somasundaram and R. Ponraj, On Mean graphs of order 5, Journal of Decision and Mathematical Sciences 9(1-3) (004), 48-58.

International J.Math. Combin. Vol.3 (009), 97-10 Chromatic Polynomial of Smarandache ν E -Product of Graphs Khalil Paryab (Department of Mathematics of Iran University of Science and Technology, Tehran, Iran.) Ebrahim Zare (Department of Mathematics of Imamali University, Tehran, Iran.) Email: paryab@iust.ac.ir, zare.math@gmail.com Abstract: Let G 1 = (V 1, E 1), G = (V, E ) be two graphs. For a chosen edge set E E, the Smarandache ν E-product G 1 νe G of G 1, G is defined by V (G 1 νe G ) = V 1 V, E(G 1 νe G ) = {(a, b)(a, b ) a = a,(b, b ) E,or b = b,(a, a ) E 1} {(a, b)(a, b ) (a, a ) E 1 and (b, b ) E}. Particularly, if E = or E, then G 1 νe G is the Cartesian product G 1 G or strong product G 1 G of G 1 and G in graph theory. Finding the chromatic polynomial of Smarandache ν E-product of two graphs is an unsolved problem in general, even for the Cartesian product and strong product of two graphs. In this paper we determine the chromatic polynomial in the case of the Cartesian and strong product of a tree and a complete graph. Keywords: Coloring graph, Smarandache ν E-product graph, strong product graph, Cartesian product graph, chromatic polynomial. AMS(000): 05C15 1. Introduction Sabidussi and Vizing defined Graph products first time in [4] [5]. A lot of works has been done on various topics related to graph products, however there are still many open problems [3]. Generally, we can construct Smarandache ν E -product of graphs G 1 and G for E E(G ) as follows. Let G 1 = (V 1, E 1 ), G = (V, E ) be two graphs. For a chosen edge set E E, the Smarandache ν E -product G 1 νe G of G 1, G is defined by V (G 1 νe G ) = V 1 V, E(G 1 νe G ) = {(a, b)(a, b ) a = a, (b, b ) E, or b = b, (a, a ) E 1 } 1 Received July 8, 009. Accepted Aug.30, 009. {(a, b)(a, b ) (a, a ) E 1 and (b, b ) E}.