IN this paper we consider simple, finite, connected and

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1 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: -5), VOL., NO., -Eqitable Labeling for Some Star and Bistar Related Graphs S.K. Vaidya and N.H. Shah Abstract In this paper we proe that the splitting graphs of K,n and B n,n are -eqitable graphs. We also show that the shadow graph of B n,n is a -eqitable graph. Frther we proe that sqare graph of B n,n is -eqitable for n (mod ) and n (mod ) and not -eqitable for n (mod ). Index Terms -eqitable labeling, star, bistar, splitting graph, shadow graph, sqare graph. MSC Codes - 5C78. I. INTRODUCTION IN this paper we consider simple, finite, connected and ndirected graph G = (V(G), E(G)) with order p and size q. For all standard terminology and notations we follow West [9]. We will gie brief smmary of definitions which are sefl for the present inestigations. Definition. : If the ertices of the graph are assigned ales sbject to certain condition(s) then is known as graph labeling. A detailed stdy on applications of graph labeling is reported in Bloom and Golomb []. According to Beineke and Hegde [] graph labeling seres as a frontier between nmber theory and strctre of graphs. For an extensie srey on graph labeling and bibliographic references we refer to Gallian [4]. Definition. : Let G = (V(G), E(G)) be a graph. A mapping f : V(G) {,,} is called ternary ertex labeling of G and f() is called the label of the ertex of G nder f. For an edge e =, the indced edge labeling f : E(G) {,,} is gien by f (e) = f() f(). Let f (), f () and f () be the nmber of ertices of G haing labels, and respectiely nder f and let e f (),e f () and e f () be the nmber of edges haing labels, and respectiely nder f. Definition. : A ternary ertex labeling of a graph G is called a -eqitable labeling if f (i) f (j) and e f (i) e f (j) for all i,j. A graph G is -eqitable if it admits -eqitable labeling. Dr. S. K. Vaidya is a proffesor at the Department of Mathematics, Sarashtra Uniersity, Rajkot, Gjarat - 65, INDIA ( samirkaidya@yahoo.co.in) N. H. Shah is a lectrer at Goernment Polytechnic, Rajkot, Gjarat - 6, INDIA ( nira.hs@gmail.com) The concept of -eqitable labeling was introdced by Cahit [] and he proed that an Elerian graph with nmber of edges congrent to (mod 6) is not -eqitable. In the same paper he proed that C n is -eqitable if and only if n (mod 6) and all caterpillars are -eqitable. Cahit [] claimed to proe that W n is -eqitable if and only if n (mod 6) bt Yossef [8] proed that W n is -eqitable for all n 4. The -eqitable labeling in the context of ertex dplication is discssed by Vaidya et al. [5] while same athors in [6] hae inestigated -eqitable labling for some shell related graphs. Vaidya et al. [7] hae discssed - eqitablity of graphs in the context of some graph operations and proed that the shadow and middle graph of cycle C n, path P n are -eqitable. Generally there are three types of problems that can be considered in this area. ) How -eqitability is affected nder arios graph operations? ) Constrct new families of -eqitable graph by inestigating sitable labeling. ) Gien a graph theoretic property P, characterize the class of graphs with property P that are -eqitable. The problems of second type are largely discssed while the problems of first and third types are rarely discssed and they are of great importance also. The present work is aimed to discss the problems of first kind. Definition.4 : The splitting graph of a graph G is obtained by adding to each ertex a new ertex sch that is adjacent to eery ertex that is adjacent to in G, i.e. N() = N( ). The resltant graph is denoted by S (G). Definition.5: The shadow graph D (G) of a connected graph G is constrcted by taking two copies of G say G and G. Join each ertex in G to the neighbors of the corresponding ertex in G. Definition.6: For a simple connected graph G the sqare of graph G is denoted by G and defined as the graph with the same ertex set as of G and two ertices are adjacent in G if they are at a distance or apart in G.

2 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: -5), VOL., NO., 4 II. MAIN RESULTS Theorem. : S (K,n ) is -eqitable graph. Proof : Let,,,..., n be the pendant ertices and be the apex ertex of K,n and,,,,..., n are added ertices corresponding to,,,,..., n to obtain S (K,n ). Let G be the graph S (K,n ) then V(G) = n+ and E(G) = n. To definef : V(G) {,,} we consider following three cases. Case : n (mod ) f() =, f( i ) = ; i n + f ( ) n n ++i = ; i ( ) f n +i = ; i n f( i ) = ; i n f ( ) n n +i = ; i + ( f = ; i n n ++i ) Case : n (mod ) f () = f () = n + = f()+ e f () = e f () = e f () = n f() =, f( i ) = ; i k + f ( k++i ) = ; i k f ( k+i ) = ; i k + f( i ) = ; i k f ( k +i ) = ; i k + f ( k++i ) = ; i k f () = f () = n+ = f () e f () = e f () = e f () = n Case : n (mod ) Since n (mod ), n = k + some k N. f() =, f( i ) = ; i k + f ( k++i ) = ; i k f ( k++i ) = ; i k + f( i ) = ; i k f ( k+i ) = ; i k + f ( k++i ) = ; i k (n+) f () = f () = f () = e f () = e f () = e f () = n Ths in each case we hae f (i) f (j) and Hence S (K,n ) is a -eqitable graph. Illstration. : -eqitable labeling of the graph S (K,7 ) is shown in Fig Figre Theorem. : S (B n,n ) is -eqitable graph. Proof : Consider B n,n with ertex set {,, i, i, i n} where i, i are pendant ertices. In order to obtain S (B n,n ), add,, i, i ertices corresponding to,, i, i where, i n. If G = S (B n,n ) then V(G) = 4(n + ) and E(G) = 6n+. To define f : V(G) {,,} we consider following for cases. Case : n =,5 The graphs S (B, ) and S (B 5,5 ) are to be dealt separately and their -eqitable labeling is shown is Fig.. and Fig.. 6 Figre 4 Figre 4 5 5

3 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: -5), VOL., NO., 5 Case : n (mod ) Since n (mod ), n = k some k N. f( ) =, f() =, f( ) =, f( i ) = ; i n f ( ) = ; f ( +i) = ; i k f ( k++i) = ; i n k f( i ) = ; i (k ) f( k +i ) = ; i n k + f ( i ) = ; i k f ( k+i) = ; i k f ( i ) = ; i = n,n f () = f () = 4k + = f () e f () = e f () = e f () = n+ Case : n (mod ) f( ) =, f() =, f( ) =, f( i ) = ; i n f ( ) = ; f ( +i) = ; i k f ( k++i) = ; i k f( i ) = ; i k + f( k++i ) = ; i k f ( i ) = ; i k f ( k+i) = ; i k f ( i ) = ; i = n,n f () = f () = 4k + = f ()+ e f () = e f () = e f () = n+ Case 4: n (mod ) Since n (mod ), n = k + some k N {}. f( ) =, f() =, f( ) =, f( i ) = ; i n f ( ) = ; f ( +i) = ; i k + f ( k++i) = ; i k f( i ) = ; i k +4 f( k+4+i ) = ; i k f ( i ) = ; i k + f ( k++i) = ; i k f ( i ) = ; n i n f () = f () = 4(k +) = f () e f () = e f () = e f () = n+ Ths in each case we hae f (i) f (j) and Hence S (B n,n ) is a -eqitable graph. Illstration.4 : -eqitable labeling of the graph S (B 6,6 ) is shown in Fig Figre 4 Theorem.5 : D (B n,n ) is -eqitable graph. Proof : Consider two copies of B n,n. Let {,, i, i, i n} and {,, i, i, i n} be the corresponding ertex sets of each copy of B n,n. Let G be the graph D (B n,n ) then V(G) = 4(n + ) and E(G) = 4(n + ). To define f : V(G) {,,} we consider following three cases. Case : n (mod ) Since n (mod ), n = k some k N. f( ) =, f() =, f( ) =, f( i ) = ; i k + f( k++i ) = ; i k f ( ( i ) = ; ) i (k ) f (k )+i = ; i k + f( i ) = ; i k f( k +i ) = ; i k + f ( i ) = ; i f ( +i) = ; i n f () = f () = 4(n+) = f () e f () = e f () = 8n+ = e f () Case : n (mod )

4 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: -5), VOL., NO., 6 f( ) =, f() =, f( ) =, f( i ) = ; i n f ( i ) = ; i n f( i ) = ; i k f( k+i ) = ; i n k f ( i ) = ; i k + f ( k++i) = ; i k f () = f () = 4(n+)+ = f ()+ e f () = e f () = e f () = 8n+4 Case : n (mod ) Since n (mod ), n = k + some k N {}. f( ) =, f() =, f( ) =, f( i ) = ; i k + f( k++i ) = ; i k f( k++i ) = ; i k + f ( i ) = ; i n f( i ) = ; i n f( n ) = ; f( n ) = ; f ( i ) = ; i n f () = f () = f () = 4(n+) e f () = e f () = 8n+4+ = e f ()+ Ths in each case we hae f (i) f (j) and Hence D (B n,n ) is a -eqitable graph. Illstration.6 : -eqitable labeling of the graph D (B 5,5 ) is shown in Fig Proof : Consider B n,n with ertex set {,, i, i, i n} where i, i are pendant ertices. Let G be the graph B n,n then V(G) = n + and E(G) = 4n +. To define f : V(G) {,,}, we consider following two cases. Case : n (mod ) Since n (mod ), n = k some k N. f() =, f() =, f( i ) = ; i k f( k+i ) = ; i k f( i ) = ; i k f( k+i ) = ; i k f () = f () = n+ = f ()+ e f () = e f () = 4n = e f() Case : n (mod ) f() =, f() =, f( i ) = ; i k f( k+i ) = ; i k + f( i ) = ; i k f( k+i ) = ; i k + f () = f () = n+ = f () e f () = e f () = 4n+ = e f ()+ Ths in both the cases we hae f (i) f (j) and Hence Bn,n is a -eqitable graph for n (mod ) and n (mod ). Illstration.8 : shown in Fig. 6. -eqitable labeling of the graph B 7,7 is Figre 5 Theorem.7 : B n,n is -eqitable graph for n (mod ) and n (mod ) Figre 6 7

5 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: -5), VOL., NO., 7 Theorem.9 : B n,n is not a -eqitable graph for n (mod ). Proof : Let G be the graph B n,n then V(G) = n + and E(G) = 4n +. Here n (mod ) therefore n = k + for some k N. Hence V(G) = k and E(G) = 6k where k = k +. So if B n,n is -eqitable then we mst hae f () = f () = f () = k and e f () = e f () = e f () = k. In B n,n, note that each i and i ( i n) are adjacent to and both moreoer and are adjacent ertices. It is obios that any edge will hae label if it is incident to one ertex with label. Following Table. shows all possible assignments of ertex label. From the Table (colmn 6) we can obsere that the edge condition iolates in all the possible assignments. Hence B n,n is not a -eqitable graph for n (mod ). III. CONCLUDING REMARKS The graphs K,n and B n,n are -eqitable being caterpillars while we show that the splitting graphs of K,n and B n,n also admit -eqitable labeling. Ths -eqitability remains inariant for the splitting graphs of K,n and B n,n. It is also inariant for shadow graph of B n,n. Moreoer we proe that the sqare graph of B n,n is -eqitable for n (mod ) and n (mod ) while not -eqitable for n (mod ). To inestigate similar reslts for other graph families and for arios graph operations is a potential area of research. REFERENCES [] I. Cahit, On cordial and -eqitable labelings of graphs, Util. Math., ol. 7, pp , 99. [] L. W. Beineke and S. M. Hegde, Strongly mltiplicatie graphs, Discss. Math. Graph Theory, ol., pp. 6-75,. [] G. S. Bloom and S. W. Golomb, Applications of nmbered ndirected graphs, Proc. of IEEE, ol. 65, pp , 977. [4] J. A. Gallian, A dynamic srey of graph labeling, The Electronic Jornal of Combinatorics, ol. 8, #DS6,. Aailable online: [5] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, Some wheel related -eqitable graphs in the context of ertex dplication, Adances and Applications in Discrete Mathematics, ol. 4, no., pp. 7-85, 9. Aailable online: [6] S. K. Vaidya, N. A. Dani, K. K. Kanani and P. L. Vihol, Cordial and -eqitable labeling for some shell related graphs, J. Sci. Res., ol., no., pp , 9. Aailable online: 7/67 [7] S. K. Vaidya, P. L. Vihol and C. M. Barasara, -eqitable graphs in the context of some graph operations, Jornal of Applied Compter Science and Mathematics, ol, no 5, pp ,. Aailable online: [8] M. Z. Yossef, A necessary condition on k-eqitable labelings, Util. Math., ol. 64, pp. 9-95,. [9] D. B. West, Introdction to Graph Theory, Prentice-Hall of India,.

6 INTERNATIONAL JOURNAL OF MATHEMATICS AND SCIENTIFIC COMPUTING (ISSN: -5), VOL., NO., 8 TABLE I i s and i s ertex label ertex label f () f () f () e f () k k k k k k k k k k k k k k k k k k 4k k k k k k k k k k k k Where f (j) = nmber of ertices haing label j for i and i where i n and j.