Sum divisor cordial labeling for star and ladder related graphs

Transcription

1 Proyecciones Journal of Mathematics Vol. 35, N o 4, pp , December 016. Universidad Católica del Norte Antofagasta - Chile Sum divisor cordial labeling for star and ladder related graphs A. Lourdusamy St. Xavier s College (autonomous), India and F. Patrick St. Xavier s College (autonomous), India Received : April 016. Accepted : October 016 Abstract A sum divisor cordial labeling of a graph G with vertex set V is a bijection f from V to {1,,, V (G) } such that an edge uv is assigned the label 1 if divides f(u) +f(v) and 0 otherwise; and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1. A graph with a sum divisor cordial labeling is called a sum divisor cordial graph. In this paper, we prove that D (K 1,n ), S 0 (K 1,n ), D (B n,n ), DS(B n,n ), S 0 (B n,n ), S(B n,n ), <K (1) 1,n K() 1,n >, S(L n ), L n K 1, SL n, TL n, TL n K 1 and CH n are sum divisor cordial graphs. AMS Subject Classification 010 : 05C78. Keywords : Divisor cordial, sum divisor cordial.

2 438 A. Lourdusamy and Jyotindra C. Prajapati 1. Introduction All graphs considered here are simple, finite, connected and undirected. For all other standard terminology and notations we follow Harary []. A labeling of a graph is a map that carries the graph elements to the set of numbers, usually to the set of non-negative or positive integers. If the domain is the set of vertices the labeling is called vertex labeling. If the domain is the set of edges then the labeling is called edge labeling. If the labels are assigned to both vertices and edges then the labeling is called total labeling. For all detailed survey of graph labeling we refer Gallian [1]. A. Lourdusamy and F. Patrick introduced the concept of sum divisor cordial labeling in [3]. In [3, 4], sum divisor cordial labeling behaviour of several graphs like path, complete bipartite graph, bistar and some standard graphs have been investigated. In this paper, we investigate the sum divisor cordial labeling behavior of D (K 1,n ), S 0 (K 1,n ), D (B n,n ), DS(B n,n ), S 0 (B n,n ), S(B n,n ), <K (1) 1,n K() 1,n >, S(L n), L n K 1, SL n, TL n, TL n K 1 and CH n. Definition 1.1. Let G =(V (G),E(G)) be a simple graph and f : V (G) {1,,, V (G) } be a bijection. For each edge uv, assign the label 1 if (f(u)+f(v)) and the label 0 otherwise. The function f is called a sum divisor cordial labeling if e f (0) e f (1) 1. A graph which admits a sum divisor cordial labeling is called a sum divisor cordial graph. Definition 1.. [9] For every vertex v V (G), the open neighbourhood set N(v) is the set of all vertices adjacent to v in G. Definition 1.3. [8] For a graph G the splitting graph S 0 (G) of a graph G is obtained by adding a new vertex v 0 corresponding to each vertex v of G such that N(v) =N(v 0 ). Definition 1.4. [8] The shadow graph D (G) of a connected graph G is obtained by taking two copies of G, sayg 0 and G 00. Join each vertex u 0 in G 0 to the neighbours of corresponding vertex u 00 in G 00. Definition 1.5. [8] Duplication of a vertex v k by a new edge e = v 0 k v00 k in agraphg produces a new graph G 0 such that N(v 0 k ) N(v00 k )=v k. Definition 1.6. [8] Let G be the a graph with V = S 1 S S 3 S i T where each S i is a set of vertices having at least two vertices of the same degree and T = V (G) \ t i=1 S i. The degree splitting graph of G denoted by DS(G) is obtained from G by adding vertices w 1,w,...,w t and joining to each vertex of S i for 1 i t.

3 Sum divisor cordial labeling for star and ladder related graphs 439 Definition 1.7. [6] The subdivision graph S(G) is obtained from G by subdividing each edge of G with a vertex. Definition 1.8. [7] Consider two copies of graph G(wheel, star, fan and friendship) namely G 1 and G. Then the graph G 0 =<G 1 G > is the graph obtained by joining the apex vertices of G 1 and G by an edge as well as to a new vertex v 0. Definition 1.9. [6] The slanting ladder SL n is a graph obtained from two paths u 1,u,,u n and v 1,v,,v n by joining each u i with v i+1 for (1 i n 1). Definition [5] The triangular ladder TL n is a graph obtained from L n by adding the edges u i v i+1, 1 i n 1, whereu i and v i, 1 i nare the vertices of L n such that u 1,u,,u n and v 1,v,,v n are two paths of length n in the graph L n. Definition [6] The corona G 1 G of two graphs G 1 (p 1,q 1 ) and G (p,q ) is defined as the graph obtained by taking one copy of G 1 and p 1 copies of G and joining the i th vertex of G 1 with an edge to every vertex in the i th copy of G. Definition 1.1. [7] The closed helm CH n is the graph obtained from helm H n by joining each pendant vertex to form a cycle.. Main results Theorem.1. The graph S 0 (K 1,n ) is sum divisor cordial graph. Proof. Let v 1,v,,v n be the pendant vertices and v be the apex vertex of K 1,n and u, u 1,u,,u n are added vertices corresponding to v, v 1,v,,v n to obtain S 0 (K 1,n ). Let G = S 0 (K 1,n ). Then, G is of order n +andsize3n. Define f : V (G) {1,,, n +} as follows: i +ifi is even i +1ifi is even f(u) =1; f(v) =; f(v i )={ i +1ifi is odd f(u i )={ i +ifi is odd

4 440 A. Lourdusamy and Jyotindra C. Prajapati f (vu i )={ 1 if i is odd 0if i is even f (vv i )={ 0 if i is odd 1if i is even f (uv i )={ 1 if i is odd 0if i is even j k We observe that, e f (0) = 3n and e f (1) = Thus, e f (1) e f (0) 1. Hence, S 0 (K 1,n ) is sum divisor cordial graph. l 3n m. Example.. A sum divisor cordial labeling of S 0 (K 1,5 ) isshowninfigure.1 Theorem.3. The graph D (K 1,n ) is sum divisor cordial graph. Proof. Let u, u 1,u,,u n and v, v 1,v,,v n be the vertices of two copies of K 1,n.LetG = D (K 1,n ). Then V (G) ={u, v, u 1,u,,u n,v 1,v,,v n } and E(G) ={uu i,vv i,uv i,vu i : 1 i n}. Also, G is of order n + and size 4n. Define f : V (G) {1,,, n +} as follows: f(u) =1; f(v) =;

5 Sum divisor cordial labeling for star and ladder related graphs 441 f(u i )={ i +1ifi is odd i +ifi is even f(v i )={ i +ifi is odd i +1ifi is even f (uu i )={ 1 if i is odd 0if i is even f (vu i )={ 0 if i is odd 1if i is even f (vv i )={ 1 if i is odd 0if i is even f (uv i )={ 0 if i is odd 1if i is even We observe that, e f (0) = n and e f (1) = n. Thus, e f (1) e f (0) 1. Hence, D (K 1,n ) is sum divisor cordial graph. Example.4. A sum divisor cordial labeling of D (K 1,4 ) isshowninfigure. Theorem.5. The graph S 0 (B n,n ) is sum divisor cordial graph.

6 44 A. Lourdusamy and Jyotindra C. Prajapati Proof. Let u, v, u 1,u,,u n,v 1,v,,v n be the vertices of B n,n. Let u 0,v 0,u 0 i,v0 i are added vertices corresponding to u, v, u i,v i to obtain S 0 (B n,n ). Let G = S 0 (B n,n ) Then, G is of order 4n +4 and size 6n +3. Define f : V (G) {1,,, 4n +4} as follows: f(u) =1; f(v) =3; f(u 0 )=; f(v 0 )=4; f(u i )=4i +;1 i n f(u 0 i )=4i +1;1 i n f(v i )=4i +3;1 i n f(v 0 i )=4i +4;1 i n f (uv) =1; f (uv 0 )=0; f (vu 0 )=0; f (u 0 u i )=1;1 i n f (uu i )=0;1 i n f (uu 0 i )=1;1 i n f (v 0 v i )=0;1 i n f (vv i )=1;1 i n f (vv 0 i )=0;1 i n We observe that, e f (0) = 3n +ande f (1) = 3n +1. Thus, e f (1) e f (0) 1. Hence, S 0 (B n,n ) is sum divisor cordial graph. Example.6. A sum divisor cordial labeling of S 0 (B 5,5 ) isshowninfigure.3

7 Sum divisor cordial labeling for star and ladder related graphs 443 Theorem.7. The graph D (B n,n ) is sum divisor cordial graph. Proof. Let u, v, u 1,u,,u n,v 1,v,,v n be the vertices of B n,n.let G = D (B n,n ). Then V (G) ={u, v, u 0,v 0 } S {u i,v i,u 0 i,v0 i :1 i n} and E(G) ={uv, u 0 v, uv 0,u 0 v 0 } S {uu i,uu 0 i,u0 u i,u 0 u 0 i,vv i,vv 0 i,v0 v i,v 0 v 0 i :1 i n}. Also, G is of order 4n +4 and size 8n +4. Define f : V (G) {1,,, 4n +4} as follows: f(u) =1; f(v) =3; f(u 0 )=; f(v 0 )=4; f(u i )=4i +1;1 i n f(u 0 i )=4i +;1 i n f(v i )=4i +4;1 i n f(v 0 i )=4i +3;1 i n f (uv) =1; f (uv 0 )=0; f (vu 0 )=0; f (v 0 u 0 )=1; f (uu i )=1;1 i n f (u 0 u i )=0;1 i n f (u 0 u 0 i )=1;1 i n

8 444 A. Lourdusamy and Jyotindra C. Prajapati f (uu 0 i )=0;1 i n f (vv i )=0;1 i n f (v 0 v i )=1;1 i n f (v 0 v 0 i )=0;1 i n f (vv 0 i )=1;1 i n We observe that, e f (0) = 4n +ande f (1) = 4n +. Thus, e f (1) e f (0) 1. Hence, D (B n,n ) is sum divisor cordial graph. Example.8. A sum divisor cordial labeling of D (B 4,4 ) isshowninfigure.4 Theorem.9. The graph DS(B n,n ) is sum divisor cordial graph. Proof. Let G = DS(B n,n ). Let V (G) ={u, v, w 1,w } S {u i,v i :1 i n} and E(G) ={uv, uw,vw } S {uu i,vv i,u i w 1,v i w 1 :1 i n}.then, G is of order n +4andsize4n +3. Define f : V (G) {1,,, n +4} as follows: f(u) =4; f(v) =; f(w 1 )=1; f(w )=3; f(u i )=i +4;1 i n f(v i )=i +3;1 i n

9 Sum divisor cordial labeling for star and ladder related graphs 445 f (uv) =1; f (uw )=0; f (vw )=0; f (uu i )=1;1 i n f (u i w 1 )=0;1 i n f (vv i )=0;1 i n f (v i w 1 )=1;1 i n We observe that, e f (0) = n +ande f (1) = n +1. Thus, e f (1) e f (0) 1. Hence, DS(B n,n ) is sum divisor cordial graph. Example.10. A sum divisor cordial labeling of DS(B 3,3 ) is shown in Figure.5 Theorem.11. The graph S(B n,n ) is sum divisor cordial graph. Proof. Let G = S(B n,n ). Let V (G) ={u, v, w} S {u i,v i,u 0 i,v0 i :1 i n} and E(G) ={uw, wv} S {uu 0 i,u0 i u i,vv 0 i,v0 i v i :1 i n}. Then, G is of order 4n +3and size 4n +. Define f : V (G) {1,,, 4n +3} as follows: f(u) =1; f(v) =; f(w) =3; f(u i )=4i;1 i n f(u 0 i )=4i +1;1 i n

10 446 A. Lourdusamy and Jyotindra C. Prajapati f(v i )=4i +3;1 i n f(v 0 i )=4i +;1 i n f (uw) =1; f (wv) =0; f (uu 0 i )=1;1 i n f (u 0 i u i)=0;1 i n f (vv 0 i )=1;1 i n f (v 0 i v i)=0;1 i n We observe that, e f (0) = n +1ande f (1) = n +1. Thus, e f (1) e f (0) 1. Hence, S(B n,n ) is sum divisor cordial graph. Example.1. A sum divisor cordial labeling of S(B 3,3 ) isshowninfigure.6 Theorem.13. The graph <K (1) 1,n K() 1,n > is sum divisor cordial graph. Proof. Let v (1) 1,v(1),,v(1) n be the pendant vertices of K (1) 1,n and v () 1,v(),,v() n be the pendant vertices of K () 1,n apex vertices of K (1) 1,n and K() 1,n common vertex x. Let G =< K (1). Let u and v be the respectively and u, v areadjacenttoanew 1,n K() 1,n >. Then, G is of order n +3 and size n +3. Define f : V (G) {1,,, n +3} as follows: f(x) =;

11 Sum divisor cordial labeling for star and ladder related graphs 447 f(u) =1; f(v) =3; f(u i 1 )=4i;1 i n f(u i )=4i +3;1 i n f(v i 1 )=4i +1;1 i n f(v i )=4i +;1 i n f (ux) =0; f (vx) =0; f (uv) =1; f (uu i 1 )=0;1 i n f (uu i )=1;1 i n f (vv i 1 )=1;1 i n f (vv i )=0;1 i n We observe that, e f (0) = n +ande f (1) = n +1. Thus, e f (1) e f (0) 1. Hence, <K (1) 1,n K() 1,n > is sum divisor cordial graph. Example.14. A sum divisor cordial labeling of <K (1) in Figure.7 1,5 K() 1,5 > is shown Theorem.15. The graph L n K 1 is sum divisor cordial graph. Proof. Let u 1,u,,u n and v 1,v,,v n be the vertices of L n.letx i be the vertex which is attached to u i and y i be the vertex which is attached

12 448 A. Lourdusamy and Jyotindra C. Prajapati to v i. Let G = L n K 1. Then, G is of order 4n and size 5n. Define f : V (G) {1,,, 4n} as follows: f(u i 1 )=8i 5; 1 i n f(u i )=8i 3; 1 i n f(x i 1 )=8i 7; 1 i n f(x i )=8i ; 1 i n f(v i 1 )=8i 6; 1 i n f(v i )=8i 1; 1 i n f(y i 1 )=8i 4; 1 i n f(y i )=8i;1 i n f (u i 1 x i 1 )=1;1 i n f (u i x i )=0;1 i n f (u i u i+1 )=1;1 i n 1 f (v i v i+1 )=0;1 i n 1 f (u i 1 v i 1 )=0;1 i n f (u i v i )=1;1 i n f (v i 1 y i 1 )=1;1 i n f (v i y i )=0;1 i n k l m We observe that, e f (0) = and e f (1) = 5n. j 5n Thus, e f (1) e f (0) 1. Hence, L n K 1 is sum divisor cordial graph. Example.16. A sum divisor cordial labeling of L 6 K 1 is shown in Figure.8

13 Sum divisor cordial labeling for star and ladder related graphs 449 Theorem.17. The graph S(L n ) is sum divisor cordial graph. Proof. Let u 1,u,,u n and v 1,v,,v n be the vertices of L n.letv 0 i be the newly added vertex between v i and v i+1.letu 0 i be the newly added vertex between u i and u i+1.letw i be the newly added vertex between v i and u i.letg = S(L n ). Then, G is of order 5n andsize6n 4. Define f : V (G) {1,,, 5n } as follows: f(v 1 )=1; f(v i+1 )=5i;1 i n 1 f(u 1 )=; f(u i+1 )=5i +3;1 i n 1 f(w 1 )=3; f(w i+1 )=5i +;1 i n 1 f(v 0 i )=5i 1; 1 i n 1 f(u 0 i )=5i +1;1 i n 1 f (v 1 v 0 1 )=0; f (v i v 0 i )=1; i n 1 f (v 0 i v i+1) =0;1 i n 1 f (u 1 u 0 1 )=1; f (u i u 0 i )=0; i n 1 f (u 0 i u i+1) =1;1 i n 1 f (v i w i )=1;1 i n

14 450 A. Lourdusamy and Jyotindra C. Prajapati f (u i w i )=0;1 i n We observe that, e f (0) = 3n ande f (1) = 3n. Thus, e f (1) e f (0) 1. Hence, S(L n ) is sum divisor cordial graph. Example.18. A sum divisor cordial labeling of S(L 5 ) isshowninfigure.9 Theorem.19. The slanting ladder SL n is sum divisor cordial graph. Proof. Let G = SL n. Let V (G) ={u i,v i :1 i n} and E(G) = {v i v i+1,u i u i+1,u i v i+1 :1 i n 1}. Then, G is of order n and size 3n 3. Define f : V (G) {1,,, n} as follows: f(u i )={ iif i 1, 0(mod 4) i 1if i, 3(mod 4) f(v i )={ i 1if i 1, 0(mod 4) iif i, 3(mod 4) f (u i u i+1 )={ 0 if i is odd 1if i is even f (v i v i+1 )={ 0 if i is odd 1if i is even f (u i v i+1 )={ 1 if i is odd 0if i is even We observe that, e f (0) = l m 3n 3 and e f (1) = j k 3n 3.

15 Sum divisor cordial labeling for star and ladder related graphs 451 Thus, e f (0) e f (1) 1. Hence, SL n is sum divisor cordial graph. Example.0. A sum divisor cordial labeling of SL 5 is shown in Figure.10 Theorem.1. The triangular ladder TL n is sum divisor cordial graph. Proof. Let G = TL n. Let V (G) ={u i,v i :1 i n} and E(G) = {u i u i+1,v i v i+1,u i v i+1 :1 i n 1} S {u i v i :1 i n}. Then, G is of order n and size 4n 3. Define f : V (G) {1,,, n} as follows: f(u i )=i 1; 1 i n f(v i )=i;1 i n f (u i u i+1 )=1;1 i n 1 f (u i v i )=0;1 i n f (v i v i+1 )=1;1 i n 1 f (u i v i+1 )=0;1 i n 1 We observe that, e f (0) = n 1ande f (1) = n. Thus, e f (0) e f (1) 1. Hence, TL n is sum divisor cordial graph. Example.. A sum divisor cordial labeling of TL 5 is shown in Figure.11

16 45 A. Lourdusamy and Jyotindra C. Prajapati Theorem.3. The graph TL n K 1 is sum divisor cordial graph. Proof. Let u 1,u,,u n and v 1,v,,v n be the vertices of TL n.let G = TL n K 1. Then V (G) ={u i,v i,x i,y i :1 i n} and E(G) = {u i u i+1,v i v i+1,u i v i+1 :1 i n 1} S {u i x i,v i y i,u i v i :1 i n}. Then, G is of order 4n and size 6n 3. Define f : V (G) {1,,, 4n} as follows: f(u i )={ 4 i 1if i is odd 4i 3if i is even f(x i )={ 4 i 3if i is odd 4i if i is even f(v i )={ 4 i if i is odd 4i 1if i is even f(y i )=4i;1 i n f (u i x i )={ 1 if i is odd 0if i is even f (v i y i )={ 1 if i is odd 0if i is even f (u i v i )={ 0 if i is odd 1if i is even f (u i v i+1 )={ 1 if i is odd 0if i is even

17 Sum divisor cordial labeling for star and ladder related graphs 453 f (u i u i+1 )=1;1 i n 1 f (v i v i+1 )=0;1 i n 1 We observe that, e f (0) = 3n ande f (1) = 3n 1. Thus, e f (1) e f (0) 1. Hence, TL n K 1 is sum divisor cordial graph. Example.4. A sum divisor cordial labeling of TL 6 K 1 is shown in Figure.1 Theorem.5. The closed helm graph CH n is sum divisor cordial graph. Proof. Let G = CH n. Let V (G) ={v, v 1,v,,v n,u 1,u,,u n } and E(G) = {vv i,u i v i : 1 i n} S {v i v i+1,u i u i+1 : 1 i n 1} S {v 1 v n,u 1 u n }. Then, G is of order n +1 and size 4n. Define f : V (G) {1,,, n +1} as follows: f(v) =1; f(v i )=i;1 i n f(u i )=i +1;1 i n f (vv i )=0;1 i n f (v i v i+1 )=1;1 i n 1 f (v 1 v n )=1; f (u i v i )=0;1 i n f (u i u i+1 )=1;1 i n 1 f (u 1 u n )=1;

18 454 A. Lourdusamy and Jyotindra C. Prajapati We observe that, e f (0) = n and e f (1) = n. Thus, e f (1) e f (0) 1. Hence, CH n is sum divisor cordial graph. Example.6. A sum divisor cordial labeling of CH 4 isshowninfigure conclusion It is very interesting and challenging as well as to investigate graph families which admit sum divisor cordial labeling. Here we have proved D (K 1,n ), S 0 (K 1,n ), D (B n,n ), DS(B n,n ), S 0 (B n,n ), S(B n,n ), <K (1) 1,n K() 1,n >, S(L n), L n K 1, SL n, TL n, TL n K 1 and CH n are sum divisor cordial graphs. References [1] J. A. Gallian, A Dyamic Survey of Graph Labeling, The Electronic J. Combin., 18 (015) # DS6. [] F. Harary, Graph Theory, Addison-wesley, Reading, Mass, (197). [3] A. Lourdusamy and F. Patrick, Sum Divisor Cordial Graphs, Proyecciones Journal of Mathematics, 35(1), pp , (016).

19 Sum divisor cordial labeling for star and ladder related graphs 455 [4] A. Lourdusamy and F. Patrick, Sum Divisor Cordial Labeling for Path and Cycle Related Graphs (submitted for publication) [5] S. S. Sandhya, S. Somasundaram and S. Anusa Some New Results on Root Square Mean Labeling, International Journal of Mathematical Archive, 5(1), pp , (014). [6] M. Seenivasan Some New Labeling Concepts, PhD thesis, Manonmaniam Sundaranar University, India, (013). [7] S. K. Vaidya and C. M. Barasara, Product Cordial Graphs in the Context of Some Graph Operations, International Journal of Mathematics and Scientific Computing, 1(), pp , (011). [8] S. K. Vaidya and N. H. Shah, Some Star and Bistar Related Cordial Graphs, Annals of Pure and Applied Mathematics, 3(1), pp , (013). [9] S. K. Vaidya and N. J. Kothari, Line Gracefulness of Some Path Related Graphs, International Journal of Mathematics and Scientific Computing, 4(1), pp , (014). A. Lourdusamy Department of Mathematics, St. Xavier s College (autonomous), Palayamkottai-6700, Tamilnadu, India lourdusamy15@gmail.com and F. Patrick Department of Mathematics, St. Xavier s College (autonomous), Palayamkottai-6700, Tamilnadu, India patrick881990@gmail.com