International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 34 ISSN 2229-5518 Some Polygonal Sum Labeling of Bistar 1.Dr.K.Amuthavalli, 2. S.Dineshkumar, ABSTRACT- A (p, q) graph G is said to admit a polygonal sum labeling if its vertices can be labeled by non -negative integers such that the induced edge labels obtained by the sum of the labels of end vertices are the first q polygonal numbers. A graph G which admits a polygonal sum labeling is called a polygonal sum graph. In this paper we have proved that the Bistar (Bn,n) admit Pentagonal, Hexagonal, Heptagonal, Octagonal, Nonagonal and Decagonal Sum Labeling. Keywords: Polygonal sum labeling and polygonal sum graph. —————————— —————————— 1. INTRODUCTION A pentagonal sum labeling of a graph G is one to one + All graphs in this paper are finite, simple and undirected. function f: V(G)→ N that induces a bijection f : E(G) → + Terms not defined here are used in the sense of Harary [2]. {A1, A2, . , Aq} of the edges of G defined by f (uv) = A graph labeling is an assignment of integers to the vertices f(u) + f(v) for every e = uv є E(G). The graph which admits or edges or both subjects to certain conditions. If the such labeling is called a pentagonal sum graph. domain of the mapping is the set of vertices (or edges) then 2.4 Definition the labeling is called a vertex labeling (or an edge labeling). Hexagonal numbers are numbers that create a hexagon. In Graph labeling was first introduced in the late 1960’s. other words 1, 6, 15, 28, 45, 66, 91, 120, 153, 190, 231, Many studies in graph labeling refer to Rosa’s research in 276, . are hexagonal numbers. 1967 [5]. In the recent years, dozens of graph labeling th techniques have been studied in over 1500 papers [3]. The n hexagonal number is denoted by Bn, then Triangular sum labeling was discussed in [6]. In [4] some Bn = nn()21 − . polygonal sum labeling of paths were discussed. In this paper we have proved that the graph B admit Pentagonal, n, n 2.5 Definition Hexagonal, Heptagonal, Octagonal, Nonagonal and A hexagonal sum labeling of a graph G is a one-to-one Decagonal sum labeling. function f : V(G) → N that induces a bijection f+ : E(G) → {B , B , . B } of the edges of G defined by f+(uv) = 2. DEFINITIONS 1 2 q f(u) + f(v) for every e= uv є E(G). The graph which admits such labeling is called a hexagonal sum graph. 2.1 Definition Polygonal numbers are just the number of vertices 2.6 Definition formed by a certain polygon. The first number in any group Heptagonal numbers are numbers that create a heptagon. of polygonal numbers is always one or a point. The second In other words 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, number is equal to the numberIJSER of vertices of the polygon. 342, 403, 469, 540, 616, 697, . are heptagonal The third number is made by extending two of the sides of numbers. the polygon from the second polygonal number, completing the larger polygon and placing vertices and other points The nth heptagonal number is denoted by C , wherever necessary. The third polygonal number is found n nn()53− by adding all the vertices and points in the resulting. then C = . n 2 2.2 Definition 2.7 Definition Pentagonal numbers are numbers that create a pentagon. A Heptagonal sum labeling of a graph G is a one-to-one In other words 1, 5, 12, 22, 35, 51, 70, 92, 117, 145 . function f : V(G) → N that induces a bijection f+ : E(G) → + are pentagonal numbers. {C1, C2, . , Cq} of the edges of G defined by f (uv) = f(u) + f(v) for every e= uv є E(G). The graph which admits th The n pentagonal number is denoted by A n, such labeling is called a heptagonal sum graph. nn()31− then An = . 2.8 Definition 2 Octagonal numbers are numbers that create an octagon. ____________________________________________ In other words 1, 8, 21, 40, 65, 96, 133,176,225,280,341,408,481,560, 645, 736, 833, 936, • Assistant Professor, PG & Research Department of Mathematics, 1045, 1160, 1281, 1408, 1541, . are octagonal numbers. Government Arts College, Ariyalur, Tamilnadu. [email protected] • Assistant Professor, Department of Mathematics, Roever College of th The n octagonal number is denoted by Dn, then Engineering and Technology, Perambalur, Tamilnadu. [email protected] = − Dn nn()3 2. 2.9 Definition 2.3 Definition An octagonal sum labeling of a graph G is a one-to-one IJSER © 2016 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 35 ISSN 2229-5518 function f : V(G) → N that induces a bijection f+ : E(G) → + {D1, D2, . , Dq} of the edges of G defined by f (uv) = f(u) + f(v) for every e= uv є E(G). The graph which admits such labeling is called an octagonal sum graph. 2.10 Definition u1 v1 Nonagonal numbers are numbers that create a nonagon. u2 en+2 e1 In other words: 1, 9, 24,46, 75, 111, 154, 204, 261, 325, v2 e2 en+3 396, 474, 559, 651, 750, 856, 969, 1089, 1216, 1350, 1491, . 1639, 1794,1956, 2125, 2301, 2484, 2674, 2871, 3075, . u e v e n+1 n-1 e2n-1 . are nonagonal numbers. th en e2n The n nonagonal number is denoted by E , then un-1 vn-1 n nn()75− E = un vn n 2 2.11 Definition Fig.1: Ordinary labeling of Bn, n A nonagonal sum labeling of a graph G is a one-to-one function f : V(G) → N that induces a bijection f+ : E(G) → + First we label the vertices of Bn, n as follows: {E1, E2,. , Eq} of the edges of G defined by f (uv) = f(u) + Define f: V (Bn, n) →N by f(v) for every e= uv є E(G). The graph which admits such labeling is called a nonagonal sum graph. fu( )= 0, ii(3− 1) 2.12 Definition fu( )= ,1 ≤≤i n Decagonal numbers are numbers that create a decagon. In i 2 other words:1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, ()()nn++13 1 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, fv()= , 1870, 2047, 2232, 2425, 2626, 2835, . are decagonal 2 numbers. 3ii (− 1) th fv(i )=() 3 n + 4 i + ,1 ≤≤i n The n decagonal number is denoted by Fn, then 2 Fn = nn()4 − 3. Then the induced edge labels are, 2.13 Definition ii(3− 1) fe( )= , 1 ≤≤i 2 n + 1, A decagonal sum labeling of a graph G is a one-to-one i 2 function f : V(G) → N that induces a bijection f+ : E(G) → + Therefore, {F1, F2, . , Fq} of the edges of G defined by f (uv) = f(u) + + f(v) for every e= uv є E(G). The graph which admits such f() EB()nn, = {1, 5, 12, 22, 35, 51, 70, 92, 117,...} labeling is called a decagonal sum graph. IJSERTherefore, f is a pentagonal sum labeling and hence Bn, n is a pentagonal sum graph. Pentagonal 1sum labeling of B7, 7 is shown 25in Fig. 253 5 3. MAIN RESULTS Here we have proved that the graph Bn,n admit Pentagonal, Hexagonal, Heptagonal, Octagonal, Nonagonal 12 145 84 117 and Decagonal Sum Labeling. 1 5 176 12 3.1 Theorem 92 22 0 92 210 The graph Bn, n is a pentagonal sum graph for n ≥ 2 . 22 118 Proof: 35 247 Let VB()nn, ={}{} u i,1 ≤≤ in ∪ vi,1 ≤≤ in ∪{}{} u ∪ v 51 70 287 EB()nn, ={ e i,1 ≤≤ i 2 n + 1} (See Fig.1) 35 330 155 51 238 195 70 Fig.2: Pentagonal sum labeling of B7, 7 IJSER © 2016 http://www.ijser.org International Journal of Scientific & Engineering Research, Volume 7, Issue 5, May-2016 36 ISSN 2229-5518 3.2 Theorem Define f: V (Bn, n) →N The graph Bn, n is a hexagonal sum graph for n ≥ 2 . Proof: fu( )= 0, Let VB()nn, ={}{} u i,1 ≤≤ in ∪ vi,1 ≤≤ in ∪{}{} u ∪ v ii(5− 3) fu( )= ,1 ≤≤i n i 2 EB()nn, ={ e i,1 ≤≤ i 2 n + 1} (See Fig.1) ()()nn++15 2 fv()= , First we label the vertices of Bn, n as follows: 2 Define f: V (B ) →N by 5ii (− 1) n, n fv( )=() 5 n + 6 i + ,1 ≤≤i n i 2 fu( )= 0, fu( )= i (2 i − 1), 1 ≤≤ i n Then the induced edge labels are, i =++ fv()()() n 1 2 n 1, ii(3− 1) fe(i )= , 1 ≤≤i 2 n + 1, fv(i )=() 4 n + 9 i − 4, 1 ≤≤ i n 2 Therefore, Then the induced edge labels are, + f() EB()nn, = {1, 7, 18, 34, 55, 81, 112, 148, 189, 235,...} fe( )= i (2 i − 1), 1 ≤≤ i 2 n + 1, i Therefore, Therefore, f is a heptagonal sum labeling and hence Bn, n is + a heptagonal sum graph. f() EB()nn, = {1, 6, 15, 28, 45, 66, 91, 120,...} Therefore f is a hexagonal sum labeling and hence Bn, n is a Heptagonal labeling of B10, 10 is shown in Fig.4 hexagonal sum graph. Hexagonal labeling of B is shown in Fig.3 7 1 4,4 56 117 21 18 183 1 7 1 1 403 18 469 6 66 46 254 6 34 91 342 34 540 0 45 45 15 IJSER 120 55 330 55 616 75 0 286 286 15 28 697 153 81 411 108 81 783 28 112 874 Fig.3: Hexagonal labeling of B 4,4 148 189 970 497 112 235 1071 3.3 Theorem The graph Bn, n is a heptagonal sum graph for n ≥ 2 .