NEIGHBOURHOOD SUM CORDIAL LABELING OF GRAPHS

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1 NEIGHBOURHOOD SUM CORDIAL LABELING OF GRAPHS A. Muthaiyan # and G. Bhuvaneswari * Department of Mathematics, Government Arts and Science College, Veppanthattai, Perambalur - 66, Tamil Nadu, India. P.G. and Research Department of Mathematics, Govt. Arts College, Ariyalur , Tamil Nadu, India. ABSTRACT In this paper, we introduce neighbourhood sum cordial labeling of graph and investigate the neighbourhood sum cordial labeling of Path, a book with n triangular pages, K,n,n, B n,n, D (K,n ) and S (K,n ). Keywords : Neighbourhood sum cordial graph, Neighbourhood sum cordial labeling, sum cordial graph.. Introduction By a graph G, we mean a finite, connected, undirected graph without loops and multiple edges, suppose graph G is disconnected means each component of G must contain at least one edge, for terms not defined here, we refer to Harary [4]. For standard terminology and notations related to the graph labeling, we refer to Gallian [3] and number theory, we refer Burton []. The concept of cordial labeling of graph was introduced by Cahit []. In [6], Shiama investigated the sum cordial labeling of graph. Patel et al. introduced the notion of neighborhood-prime labeling of graph [5]. Motivated by the study of sum cordial labeling and neighborhood-prime labeling, we introduce neighbourhood sum cordial labeling of graph and and investigate the neighbourhood sum cordial labeling of Path, a book with n triangular pages, K,n,n, B n,n, D (K,n ) and S (K,n ). The brief summaries of definitions which are necessary for the present investigation are provided below. Definition :. A complete bipartite graph K,n is called a star and it has n+ vertices and n edges. K,n,n is the graph obtained by the subdivision of the edges of the star K,n. Definition :. The set of all vertices adjacent to a vertex v is called the neighbourhood of the vertex v and it is denoted by N(v). Definition :.3 For a graph G the splitting graph S (G) of a graph G is obtained by adding a new vertex v corresponding to each vertex v of G such that N(v) = N(v ). Definition :.4 Bistar B n,n is the graph obtained by joining the centre (apex) vertices of two copies of K,n by an edge. Definition :.5 A vertex switching G v of a graph G is obtained by taking a vertex v of G, removing the entire edges incident with v and adding edges joining v to every vertex which are not adjacent to v in G. Definition :.6 The shadow graph D (G) of a connected graph G is constructed by taking two copies of G say G and G. Join each vertex u in G to the neighbours of the corresponding vertex v in G. Vol: I. Issue XLIX, March 7 8

2 Definition :.7 A graph labeling is the assignment of unique identifiers to the edges and vertices of a graph. Definition :.8 A mapping f:v(g) {,} is called binary vertex labeling of G and f(v) is called the label of the vertex v of G under f. If for an edge e = uv, the induced edge labeling f * : E(G) {,} is given by f * (e) = f(u) f(v). Then v f (i) = number of vertices of having label i under f and e f (i) = number of edges of having label i under f *. A binary vertex labeling f of a graph G is called a cordial labeling if v f () v f () and e f () e f (). A graph G is cordial if it admits cordial labeling. Definition :.9 A binary vertex labeling of a graph G with induce edge labeling f * : E(G) {,} defined by f * (uv) = (f(u)+f(v))(mod) is called sum cordial labeling if v f () v f () and e f () e f (). A graph G is sum cordial if it admits sum cordial labeling. Definition :. Let G = (V,E) be a graph with n vertices. A bijective function f : V(G) {,,3,...,n} is said to be a neighborhood-prime labeling, if for every vertex v V(G) with deg(v) >, gcd {f(u): u N(v)} =. A graph which admits neighborhood-prime labeling is called a neighborhood-prime graph. Definition :. Let G = (V,E) be a simple graph with n vertices and f : V(G) {,, 3,..., n} be a bijection. For each edge uv, the induced edge labeling f * : E(G) {,} is given by f * (uv) = (f(u)+f(v))(mod). Then the function f is said to be a neighbourhood sum cordial labeling, if for every vertex v V(G) with deg(v) >, gcd {f(u): u N(v)} = and e f () e f (), where e f (i) = number of edges of having label i under f *. A graph G is neighbourhood sum cordial if it admits neighbourhood sum cordial labeling.. Main Theorems Theorem :. The path P n is a neighbourhood sum cordial graph for n. Proof: Let P n be a path of n vertices. Let v, v,..., v n and e, e,..., e n be the consecutive vertices and edge of the path. Let G = P n. Then V(G) = n and E(G) = n Define the vertex labeling f : V(G) {,,,n} as follows: Case (i): n is odd. n f(v i ) = i, for i, n n f(v i ) = i+, for i. Consider the vertex v i v, v n and deg(v i ) >. Then N(v i ) = {v i, v i+ } and gcd{f(x) : x N(v i )} = for all i, since f(v i ) and f(v i+ ) are consecutive integers. In view of the above labeling pattern we have, e f () = e f () = n. Hence e f () e f (). Therefore G is a neighbourhood sum cordial graph for n is odd. Case (ii): n is even. f(v i ) = i, for i n, f(v i ) = i+ n, for i n. Consider the vertex v i v, v n and deg(v i ) >. Then N(v i ) = {v i, v i+ } and gcd{f(x) : x N(v i )} = for all i, since f(v i ) and f(v i+ ) are consecutive integers. Vol: I. Issue XLIX, March 7 8

3 In view of the above labeling pattern we have, e f () = e f () + = n for n (mod4) and ef () = e f ()+ = n for n (mod4). Hence ef () e f (). Thus the vertex and edge conditions of the neighbourhood sum cordial labeling are satisfied. Therefore G is a neighbourhood sum cordial graph for n is even. Thus f admits neighbourhood sum cordial labeling of G. Hence, the path P n is a neighbourhood sum cordial graph for every n. Example :. The path P 6 and its neighbourhood sum cordial labeling are shown in figure. Theorem :. 4 Figure. Given a positive integer n, there is a neighbourhood sum cordial graph G which has n vertices. Let n be any positive integer. Case (i) : n is even. In the following manner we construct n vertices graph G. Let v, v,, v n be the vertices of G. Form a path containing n vertices v, v,..., n Define the vertex labeling f : V(G) {,,,n} as follows f(v i ) = i for i n, n v and attach vertices n v,..., v n to the vertex v. n f( v n ) = i for i. i Consider x = v. Then gcd of the labels of vertices in N(v ) is. Since the set {f(y) : y N(v )} contains the numbers and 3. Consider x = v. Then N(v ) = {v, v 3 } and gcd of the labels of vertices in N(v ) is. Since the set {f(x) : x N(v )} contains the number. Consider x = v 3,..., v n. Then N(v i ) = {v i, v i+ } for i = 3,4,, n and gcd{f(y) : y N(vi )} = for all i, since f(v i ) and f(v i+ ) are consecutive odd integers. In view of the above labeling pattern we have, e f () = e f () + = n Hence e f () e f (). Therefore, the resultant graph G is neighbourhood sum cordial for n is even. Case (ii) : Suppose n is odd. In the following manner we construct n vertices graph G. Let v, v,, v n be the vertices of G. Form a path containing n vertices v, v,..., Define f : V(G) {,,, n} as follows v n and attach n vertices v n 3, v n 3,..., v n to the vertex v. Vol: I. Issue XLIX, March 7 83

4 f(v i ) = i f( v ( n ) ) = i for i i for i n 8 n 9 Consider x = v. Then gcd of the labels of vertices in N(v ) is. Since the set {f(y) : y N(v )} contains the numbers and 3. Consider x = v. Then N(v ) = {v, v 3 } and gcd of the labels of vertices in N(v ) is. Since the set {f(x) : x N(v )} contains the number. Consider x = v 3,..., v n. Then N(v i ) = {v i, v i+ } for i = 3,, n integers. In view of the above labeling pattern we have, e f () = e f () = and gcd{f(y) : y N(vi )} = for all i, since f(v i ) and f(v i+ ) are consecutive odd n Hence e f () e f (). Therefore, the resultant graph G is neighbourhood sum cordial graph for n is odd. Therefore, given a positive integer n, there is a neighbourhood sum cordial graph G which has n vertices. Example :. The graph G with vertices and its neighbourhood sum cordial labeling is shown in figure.. G : Theorem :.3 Figure. A book with n triangular pages is a neighbourhood sum cordial graph. Let G be a book with n triangular pages. Let u and v be the vertices of the common edge of the triangular pages and let v, v,..., v n be the vertices of other ends of the triangle. Then V(G) = n+ and E(G) = n+. Define vertex labeling f : V(G) {,,..., n+}as follows. f(u) = f(v) = f(v i ) = +i for i n Consider the vertex x = u. Then the gcd of the labels of vertices in N(u) is. Since the set {f(y) : y N(u)} contains consecutive labels from,3,,n. Consider the vertex x = v. Then the gcd of the labels of vertices in N(v) is. Since the set {f(y) : y N(v)} contains the number. Consider the vertex x = v i for i = to n. Then the gcd of the labels of vertices in N(w i ) is. Since the set {f(y) : y N(w i ), i =,,, n} contains the number. In view of the above labeling pattern we have, e f () = e f () + = n+. Hence e f () e f (). Therefore a book with n triangular pages is a neighbourhood sum cordial graph. Vol: I. Issue XLIX, March 7 84

5 Example :.3 The book graph with 5 triangular pages and its neighbourhood sum cordial labeling are shown in figure Figure.3 Theorem :.4 The graph K,n,n is neighbourhood sum cordial graph. Proof Let G be a K,n,n. Let V(G) = {v, v i, u i : i n} and E(G) = {vv i,v i u i : i n}. Then V(G ) = n+ and E(G) = n. Define vertex labeling f : V(G) {,,,n+} as follows. f(v) =, f(v i ) = i+, for i n f(u i ) = i, for i n Consider the vertex x = v. Then the gcd of the labels of vertices in N(x) is. Since the label of vertices in N(x) are odd consecutive integers. Consider the vertex x = v i for i n. Then the gcd of the labels of vertices in N(x) is. Since one of the label of vertices in N(x) is. In view of the above labeling pattern we have, e f () = e f () = n for any n. Hence e f () e f (). Hence, the graph K,n,n is neighbourhood sum cordial graph for any n, n. Example.4 The graph K,6,6 and its neighbourhood sum cordial labeling is shown in figure.4. Theorem Vol: I. Issue XLIX, March Figure.4 The graph B n,n is neighbourhood sum cordial graph, where n. Let B n,n be a graph with vertex set {u,v,u i,v i, i n} and edge set {uv, uu i, vv i, : i n}, where u i, v i are pendant vertices. Let G be the graph B n,n. Then V(G) = n+ and E(G) = n+. Define vertex labeling f : V(G) {,,,n+} as follows. f(u) =,

6 f(v) =, f(u i ) = i+, for i n f(v i ) = n++i, for i n Consider the vertex x = u, v. Then the gcd of the labels of vertices in N(x) is. Since the label of vertices in N(x) has consecutive integers. In view of the above labeling pattern we have, e f () = e f () + = n+ if n is even or e f () = e f () + = n+ if n is odd. Hence e f () e f (), if n is odd or even. Therefore B n,n is a neighbourhood sum cordial graph for n. Example.5 The graph B 4,4 and its neighbourhood sum cordial labeling is shown in figure Figure.5 7 Theorem :.6 D (K,n ) is neighbourhood sum cordial graph, where n. Let v, v, v,, v n be the vertices of K,n. Consider two copies of K,n. Let v, v, v,, v n be the vertices of the first copy of K,n and w, w, w,, w n be the vertices of the second copy of K,n where v and w are the respective apex vertices. Let G be D (K,n ). Then V(G) = n+ and E(G) = 4n. Define vertex labeling f : V(G) {,,,n+} as follows. f(v) =, f(w) =, f(v i ) = +i for i n f(w i ) = n++i, for i n Consider the vertex x = v or w. Then the gcd of the labels of vertices in N(x) is. Since the label of vertices in N(x) have consecutive integers. Consider the vertex x = v i or w i, for i n. Then the gcd of the labels of vertices in N(x) is. Since the label of vertices in N(x) consists of and. In view of the above labeling pattern we have, e f () = e f () = n for any n. Hence e f () e f (). Thus f admits neighbourhood sum cordial labeling of G. Hence, the D (K,n ) is neighbourhood sum cordial graph, where n. Vol: I. Issue XLIX, March 7 86

7 Example.6 The graph D (K,4 ) and its neighbourhood sum cordial labeling is shown in figure Theorem :.7 7 Figure.6 The graph S (K,n ) is neighbourhood sum cordial graph, where n. Let v,v,v 3,,v n be the pendant vertices and v be the apex vertex of K,n and u, u, u, u 3,, u n are added vertices corresponding to v, v, v, v 3,, v n to obtain the graph S (K,n ). Let G be the graph S (K,n ). Then v, v, v, v 3,, v n, u, u, u, u 3,, u n are vertices of G and vv i, uv i, vu i, uu i, i n are edges of G. Then V(G) = n+ and E(G) = 3n. Define vertex labeling f : V(G) {,,,n+} as follows. f(v) =, f(v i ) = +i for i n f(u) =, f(u i ) = n++i, for i n Consider the vertex x = v, u. Then the gcd of the labels of vertices in N(x) is. Since the label of vertices in N(x) has consecutive integers. Consider the vertex x = u i, for i n. Then the gcd of the labels of vertices in N(x) is. Since the label of vertices in N(x) are the numbers and. In view of the above labeling pattern we have, 3n e f () = e f () = for any n. 8 9 Hence e f () e f (). Thus f admits neighbourhood sum cordial labeling of G. Hence, the graph D (K,n ) is neighbourhood sum cordial graph, where n. Vol: I. Issue XLIX, March 7 87

8 Example.7 The graph S (K,4 ) and its neighbourhood sum cordial labeling is given in figure Figure Conclusions In this paper, the neighbourhood sum cordial labeling of graph is introduced and the neighbourhood sum cordial labeling of path, a book with n triangular pages, K,n,n, B n,n, D (K,n ) and S (K,n ) are presented. References. I. Cahit, Cordial graphs: A weaker version of graceful and harmonious graphs, Ars Combinatoria, 3, pp. -7, David M. Burton, Elementary Number Theory, Second Edition, Wm. C. Brown Company Publishers, J. A. Gallian, A dynamic survey of graph labeling, The Electronic Journal of Combinatorics, 6, # DS6, F. Harary, Graph Theory, Addison-Wesley, Reading, Mass, S.K. Patel and N.P. Shrimali, Neighborhood-prime labeling, International Journal of Mathematics and Soft Computing, Vol.5, No., 5, pp J. Shiama, Sum cordial labeling for some graphs, Int. J. Math. Archive 3,, pp Vol: I. Issue XLIX, March 7 88