Square Difference Prime Labeling for Some Snake Graphs
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1 Global Journal of Pure and Applied Mathematics. ISSN Volume 13, Number 3 (017), pp Research India Publications Square Difference Prime Labeling for Some Snake Graphs Sunoj B.S* and Mathew Varkey T.K.** *Department of Mathematics, Government Polytechnic College, Attingal , Kerala, India. **Department of Mathematics, TKM College of Engineering, Kollam , Kerala, India. Abstract A graph labeling is an assignment of integers to the vertices or edges or both subject to certain conditions. The graph for which every edge(uv), the labels assigned to u and v are whole numbers and for each vertex of degree at least, the g c d of the labels of the incident edges is 1. Here we characterize some snake graphs for square difference prime labeling. Keywords: Graph labeling, prime labeling, square difference, prime graphs, snake graphs. 1. INTRODUCTION All graphs in this paper are finite and undirected. The symbol V(G) and E(G) denotes the vertex set and edge set of a graph G. The graph whose cardinality of the vertex set is called the order of G, denoted by p and the cardinality of the edge set is called the size of the graph G, denoted by q. A graph with p vertices and q edges is called a (p,q)- graph.
2 1084 Sunoj B.S. and Mathew Varkey T.K. A graph labeling is an assignment of integers to the vertices or edges. Some basic notations and definitions are taken from [1],[],[3] and [4]. Some basic concepts are taken from Frank Harary [1]. In this paper we investigated square difference prime labeling of some snake graphs.. MAIN RESULTS Definition.1 Let G = (V(G),E(G)) be a graph with p vertices and q edges. Define a bijection f : V(G) {0,1,, , p-1} by f(vi) = i 1, for every i from 1 to p and define a 1-1 mapping f sdp : E(G) set of natural numbers N by f sdp (uv) = f(u) - f(v). The induced function f sdp is said to be a square difference prime labeling, if for each vertex of degree at least, the g c d of the labels of the incident edges is 1. Definition. A graph which admits square difference prime labeling is called a square difference prime graph. Definition.3 The triangular snake Tn is obtained from the path Pn by replacing each edge of the path by a triangle C3. Definition.4 An alternate triangular snake A(Tn) is obtained from a path u1,u,----,un by joining uiui+1( alternately) to a new vertex vi. Definition.5 A double triangular snake D(Tn) consist of two triangular snakes that have a common path. Theorem:.1 An alternate triangular snake A(Tn) admits square difference prime labeling, when n is a positive even integer and the triangle starts from the first vertex. Proof : Let G = A(Tn) and let v1,v, ,v3n/ are the vertices of G. Here V(G) = 3n and E(G) = n 1.
3 Square Difference Prime Labeling for Some Snake Graphs 1085 Define a function f : V {0,1,, , 3n -1} by f(vi) = i-1, i = 1,,------, 3n f sdp (v i v i+1 ) = i-1, i = 1,,3, , 3n -1 f sdp (v 3i v 3i ) = 1i-8, i = 1,,3, , n According to this pattern A(Tn), admits square difference prime labeling. Example.1 G = A(T6) Fig - i Theorem:. An alternate triangular snake A(Tn) admits square difference prime labeling, when n is a positive even integer and the triangle starts from the second vertex. Proof : Let G = A(Tn) and let v1,v, ,v3n-/ are the vertices of G. Here V(G) = 3n and E(G) = n 3. Define a function f : V {0,1,, , 3n -1} by f(vi) = i-1, i = 1,,------, 3n f sdp (v i v i+1 ) = i-1, i = 1,,3, , 3n 4 f sdp (v 3i 1 v 3i+1 ) = 1i-4, i = 1,,3, , n
4 1086 Sunoj B.S. and Mathew Varkey T.K. According to this pattern A(Tn), admits square difference prime labeling. Example. G = A(T8) Fig - ii Theorem:.3 An alternate triangular snake A(Tn) admits square difference prime labeling, when n is a positive odd integer and the triangle starts from the first vertex. Proof : Let G = A(Tn) and let v1,v, ,v3n-1/ are the vertices of G. Here V(G) = 3n 1 and E(G) = n. Define a function f : V {0,1,, , 3n 1-1} by f(vi) = i-1, i = 1,,------, 3n 1 f sdp (v i v i+1 ) = i-1, i = 1,,3, , 3n 3 f sdp (v 3i v 3i ) = 1i-8, i = 1,,3, , n 1 According to this pattern A(Tn), admits square difference prime labeling. Example.3 G = A(T7) Fig - iii
5 Square Difference Prime Labeling for Some Snake Graphs 1087 Theorem:.4 An alternate triangular snake A(Tn) admits square difference prime labeling, when n is a positive odd integer and the triangle starts from the second vertex. Proof : Let G = A(Tn) and let v1,v, ,v3n-1/ are the vertices of G. Here V(G) = 3n 1 and E(G) = n. Define a function f : V {0,1,, , 3n 3 } by f(vi) = i-1, i = 1,,------, 3n 1 f sdp (v i v i+1 ) = i-1, i = 1,,3, , 3n 3 f sdp (v 3i 1 v 3i+1 ) = 1i-4, i = 1,,3, , n 1 According to this pattern A(Tn), admits square difference prime labeling. Example.4 G = A(T5) Fig -iv Theorem:.5 The triangular snake S3,n admits square difference prime labeling. Proof : Let G = S3,n and let v1,v, ,vn-1 are the vertices of G. Here V(G) = n-1 and E(G) = 3n-3 Define a function f : V {0,1,, , n } by f(vi) = i-1, i = 1,,------,n-1
6 1088 Sunoj B.S. and Mathew Varkey T.K. f sdp (v i v i+1 ) = i-1, i = 1,,3, ,n- f sdp (v i 1 v i+1 ) = 8i-4, i = 1,,3, , n-1 According to this pattern S3,n admits square difference prime labeling,. Example :.5 G = S3,8 Fig - v Theorem:.6 The double triangular snake D(Tn) admits square difference prime labeling. Proof : Let G = D(Tn) and let v1,v, ,v3n- are the vertices of G. Here V(G) = 3n- and E(G) = 5n-5 Define a function f : V {0,1,, , 3n 3} by f(vi) = i-1, i = 1,,------,3n- f sdp (v 3i v 3i+1 ) = 18i-9, i = 1,,3, ,n-1 f sdp (v 3i v 3i ) = 1i-8, i = 1,,3, ,n-1 f sdp (v 3i v 3i 1 ) = 6i-5, i = 1,,3, ,n-1 f sdp (v 3i v 3i+1 ) = 6i-1, i = 1,,3, ,n-1 f sdp (v 3i 1 v 3i+1 ) = 1i-4, i = 1,,3, ,n-1 According to this pattern D(Tn) admits square difference prime labeling,.
7 Square Difference Prime Labeling for Some Snake Graphs 1089 Example :.6 G = D(T6) Fig -vi REFERENCES [1] F Harary, Graph Theory, Addison-Wesley,Reading, Mass, (197) [] Joseph A Gallian, A Dynamic Survey of Graph Labeling, The Electronic Journal of Combinatorics(015), #DS6, pp [3] S.Uma Maheswari, Dr. K Indirani, On Triangular Sum Prime Graphs, International Journal of Computer and Mathematical Sciences, Volume 5, Issue 3 March 016, pages 0- [4] T K Mathew Varkey, Some Graph Theoretic Generations Associated with Graph Labeling, PhD Thesis, University of Kerala 000.
8 1090 Sunoj B.S. and Mathew Varkey T.K.
Sunoj B S *, Mathew Varkey T K Department of Mathematics, Government Polytechnic College, Attingal, Kerala, India
International Journal of Scientific Research in Computer Science, Engineering and Information Technology 2018 IJSRCSEIT Volume 3 Issue 1 ISSN : 256-3307 Mean Sum Square Prime Labeling of Some Snake Graphs
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