Square Difference Labeling Of Some Path, Fan and Gear Graphs

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1 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 ISSN Square Dfferece Labelg Of Some Path, Fa ad Gear Graphs J.Shama Assstat Professor Departmet of Mathematcs CMS College of Scece ad Commerce Combatore , Taml Nadu, Ida Emal: Abstract: I ths paper t s proved that some ew graphs admt square dfferece labelg. A fucto f s called a square dfferece labelg f there exst a bjecto f: V(G) {0,,,.p-} such that the duced fucto f * :E(G) N gve by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. Ay graphs whch satsfes the Square dfferece labelg s called the Square dfferece graph. Here t s vestgated that the cetral graphs of path, square graphs of path, some path related graphs, fa ad gear graphs are all square dfferece graphs. Keywords: Square dfferece labelg, square dfferece graphs, cetral graphs ad power graphs.. INTRODUTION All graphs ths paper are fte ad udrected. For all other termology ad otatos I follow Harary [ ].Let G (V,E) be a graph where the symbols V (G) ad E (G) deotes the vertex set ad the edge set. The cardalty of the vertex set s called the order of the graph G ad t s deoted by p. The cardalty of the edge set s called the sze of the graph G ad t s deoted by q. Hece the graph s deoted by G (p, q).if the vertces of the graph are assged values subject to certa codtos s kow as graph labelg. The deftos for power graphs are used from Gary Chatrad [4].Some basc cocepts are take from [5] ad [] A dyamc survey o graph labelg s regularly updated by Galla [3] ad t s publshed by Electroc Joural of Combatorcs. The square sum labelg s prevously defed by V.Ajtha, S.Arumugam ad K.A.Germa []. I proved some mddle ad total graphs [6] are square sum graphs. The cocept of square dfferece labelg was frst troduced by J.Shama [9]. Myself proved [9] ad [0] may stadard graphs lke P, C, complete graphs, cycle cactus, ladder, lattce grds, quadrlateral sakes, Wheels, K+ m K, comb, star graphs, m K3, m C3,duplcato of vertces by a edge to some star graphs ad crow graphs are square dfferece graphs. Also proved that the path s a odd square dfferece graphs ad star graphs are perfect square graphs. Some IJSER 03

2 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 ISSN graphs lke shadow ad splt graphs [7], [8] ca also be vestgated for the square dfferece labelg. Result-: Path graphs are square dfferece graphs [0] Result-: Wheels graphs are square dfferece graphs [9]. MAIN RESULTS Defto.: Let G = (V(G),E(G)) be a graph.a fucto f s called a square dfferece labelg f there exst a bjecto f: V(G) {0,,,.p-} such that the duced fucto f * :E(G) N gve by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. Defto.: Ay graph whch admts square dfferece labelg s called square dfferece graph. Defto.3:The k th power G k of a coected graph G,where k, s that graph wth V(G k ) = V(G) for whch uv E(G k ) f ad oly f dg (u,v) k.the graphs G ad G 3 are also referred to as square ad cube respectvely of G. Defto.4: Let G be a fte udrected graph wth o loops ad multple edges. The cetral graph C(G) of a graph G s obtaed by subdvdg each edge of G exactly oce ad jog all the o adjacet vertces of G. By defto vertces of the cetral graph s Pc(G) = p + q, where p ad q are umber of vertces ad edges G. For ay graph (p,q) there exsts p vertces of degree p- ad q vertces of degree C(G) Defto.5: The graph (P; K ) s obtaed from a path P by jog a pedat edge at each vertex of the path. Defto.6: Let Sm be a star graph wth vertces v, w, w,..., wm. Defe (P; Sm ) the graph obtaed from m copes of Sm ad the path P: u, u,..., u by jog uj wth the vertex v of the j th copy of Sm by meas of a edge, m 3, j. Theorem-.7: Cetral graph of a path P admts square dfferece labelg Proof: Let P : u, u,, u. Sub dvde the edges by v, v,, v-. Let G = C(P) p = V (G) = - (+)/ ad q = E (G) = (-) Defe f: V (G) {0,,,.p-} as follows f(u) = -, - f(v) = -, - We costruct the vertex labeled sets as follows. V = { f ( )} u = { } = {0,,4,,, -} V = { f ( )} v = { j } = {, 3, 5,, -3} We costruct the edge labeled sets as follows. { * E = f ( u v )} = { ( ) ( ) IJSER 03

3 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 3 ISSN = = { 4 3 {4 3} = {,5,9,...,4-7} { E = f *( v u )} = { [ f ( v )] [ f ( u )] = {( ) () = { [ f ( u )] [ f ( u )] 3 = {( ) ( 4) = { 4 = { 36, 60,...,4+} All the edges all the sets are dstct. Hece the cetral graphs of path admt square dfferece labelg. Example-.8: Cetral graph of a path P4 admts square dfferece labelg = = { 4 {4 } = {3,7,,...,4-5} { E3 = f *( u u )} = { [ f ( u )] [ f ( u )] = {( ) ( ) {6 = } ={ 6,3,... } { 3 E4 = f *( u u )} 3. Some path related graphs Theorem- 3.: The graph P s a square dfferece graph. Proof: Let P: u, u,..., u be a path. Let G = P p = V (G) = ad q = E (G) = -3. Defe a vertex labelg f: V (G) {0,,,...p-} by f(u) = -, ad the duced edge labelg fucto f * : E (G) N defed by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. such that f * (e) f * (ej) for every e ej IJSER 03

4 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 4 ISSN E = {(uu+) / -} E = {(uu+) / -} I E f * (u u+) = { [ f ( u )] [ f ( u )] I E = {( ) = { ={,3,5,... -3} f * (u u+) = { [ f ( u )] [ f ( u )] = { ( ) ( ) = { 4 ={4,8,,... 4(-)} Clearly the labelg of edges of E ad that of E are all dstct, as E cotas odd umbers ad E cotas eve umbers.therefore P are square dfferece graphs. Example 3.: The graph P6 s a square dfferece graph Proof: Let G = (P, K ), V (G) = ad E (G) = -. Defe the vertex labelg f: V (G) {0,,,.p-} by f(u) = -, f(v) = + -, ad the duced edge labelg fucto f * : E (G) N defed by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. such that f * (e) f * (ej) for every e ej E = {(uu+) / -} E = {(uv) / } I E f * (u u+) = { [ f ( u )] [ f ( u )] = {( ( ) = {( ( ) ={,3,5,... -3} I E f * (u v) = { [ f ( u )] [ f ( v )] Theorem- 3.3: The graph (P, K ) s a square dfferece graph = {( ( ) ( ) IJSER 03

5 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 5 ISSN = {( (( ) ={,+,4+, } Clearly the edge labels are dstct. Hece the graphs (P, K ) admt the square dfferece labelg. Example 3.4: The graph (P, K ) s a square dfferece graph E = {(uv) / } E3 = {(vw) / } I E f * (u u + ) = { [ f ( u )] [ f ( u )] = {( ) ( ) = {( ) ={,3,5,... -3} I E Theorem- 3.5: The graph (P, S ) s a square dfferece graph Proof: Let G = (P, S ), V (G) = 3 ad E (G) = 3-. Defe the vertex labelg f: V (G) {0,,,.p-} by f(u) = -, f(v) = +-, f(w) = + -, ad the duced edge labelg fucto f * : E (G) N defed by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. such that f * (e) f * (ej) for every e ej E = {(uu+) / -} f * (u v ) = { [ f ( u )] [ f ( v )] I E3 = {( ( ) ( ) = {( ( ={,+,4+, } f * (v w) = { [ f ( v )] [ f ( w )] = { ( ) ( ) = {( ( 3 ={3,+3,4+3, } Clearly the edge labels are dstct. Hece the graphs (P, S ) are square dfferece graphs. IJSER 03

6 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 6 ISSN Example 3.6: The graph (P4, S ) s a square dfferece graph = {( ( ) = {( ={,3,5,... -3} Theorem- 3.7: The graph (P, S ) s a square dfferece graph Proof: Let G = (P, S ), V (G) = 4 ad E (G) = 4-. Defe the vertex labelg f: V (G) {0,,,.p-} by f(u) = -, f(v) = - +, f(w) =- +, ad the duced edge labelg fucto f * : E (G) N defed by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. such that f * (e) f * (ej) for every e ej E = {(uu+) / -} E = {(uv) / } E3 = {(vw -) / } E4 = {(vw ) / } I E f * (u v) = { [ f ( u )] [ f ( v )] I E3 = {( ( ) ( ) = {( ( ={,+,4+, } f * (v w-) = { [ f ( v )] [ f ( w )] I E4 = { ( ) ( ) = {( 3( = {3, 3(+),..., 3(-) } f * (v w) = { [ f ( v )] [ f ( w )] I E f * (u u+) = { [ f ( u )] [ f ( u )] = { ( ) ( ) = {( ( IJSER 03

7 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 7 ISSN = {( 3( ) ( All the edge labels E, E, E3 ad E4 are all dstct. Hece the graphs (P, S ) are square dfferece graphs. Example 3.8: The graph (P4, S ) s a square dfferece graph ad the duced edge labelg fucto f * : E (G) N by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. E = {(uu+) / -} E = {(cu) / } The vertex sets are V= f(c) = 0 V f ( u ) Theorem- 3.9: The graph (P, S3 ) s a square dfferece graph Proof9 Example 3.0: The graph (P4, S3 ) s a square dfferece graph ( ) = {,,...,-} I E * E = { f ( cu )} { [ f ( c)] [ f ( u )] 4. Some other graphs Theorem- 4.: Fa graphs are square dfferece graphs. Proof: Let F be a fa wth vertces u,u,u,...,u. Let G = F, V (G) = + ad E (G) = 3. Defe the vertex labelg f: V (G) {0,,,.p-} by f(u) =, ad f(c) = 0 0 ( ) ( ) = {,9,5,...,(-) } I E {,3 E = f * ( u u )},,3 { [ f ( u )] [ f ( u )] IJSER 03

8 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 8 ISSN = { ( ) ( ),3 =,3 { ={3,7,,...,4-} clearly the edge labels are dstct. Hece the Fas are all square dfferece graphs. Example 4.: The fa graph F5 s a square dfferece graph E = {(uv) / } E = {(vu+) / } E3 = {(uu) / } I E f * (u v) = { [ f ( u )] [ f ( v )] = {( ) ( ) = {( ( ={48,60,7,... 3 } I E f * (v u+) = { [ f ( v )] [ f ( u )] Theorem- 4.3: Gear graphs G are square dfferece graphs. Proof: Let V (G) = + ad E (G) = 3. Let us defe the vertex labelg f: V (G) {0,,,.p-} as follows f(u) = 0 f(u) =, = {( ) ( ) = {( ) = {45, 55,... } * E3 = { f ( u u )} f(v) = +, Ad the duced edge labelg fucto { [ f ( u )] [ f ( u )] f * : E (G) N defed by f * (u v) = [f(u)] - [f(v)] for every uv E(G) are all dstct. such that f * (e) f * (ej) for every e ej = { (0) ( ) =, { IJSER 03

9 Iteratoal Joural of Scetfc & Egeerg Research Volume 4, Issue3, March-03 9 ISSN ={,4,,..., } I E4 * E4 = { f ( u v )} { [ f ( u )] [ f ( v = { () ={4 -} All the edges are dstct. Hece all the gear graphs G are square dfferece graphs. Example 4.4: The gear graph G6 s a square dfferece graph )] [] Frak Harary, Graph theory, Narosa Publshg House- (00). [3] J A Galla, A dyamc survey of graph labelg, The Electrocs joural of Combatores, 7(00) # DS6 [4]Gary Chartrad, ad Lesak, Graphs ad dgraphs, fourth edto, Chapma & HALL/CRC [5]Gary Chartrad, Pg Zhag, Itroducto to Graph theory, McGraw- Hll Iteratoal Edto [6] J.Shama Square sum labelg for some mddle ad total graphs Iteratoal Joural of Computer Applcatos ( ) Volume 37- No.4 Jauary 0 [7] J.Shama Square dfferece labelg for some graphs Iteratoal Joural of Computer Applcatos ( ) Volume 44- No.4, Aprl 0 [8] J.Shama Some Specal types of Square dfferece graphs Iteratoal Joural of Mathematcal archves- 3(6),0, ISSN [9]D B West, Itroducto to Graph Theory, Pretce-Hall, Ida, 00 Cocluso: I ths paper t s proved that some path graphs admts square dfferece labelg. Also proved some fa ad gear graphs admt square dfferece labelg. I my prevous papers [7],[8] t s proved that some graphs are square dfferece graphs. The square dfferece labelg ca be vestgated for more graphs. It s a ope problem. REFERENCES []V.Ajtha, S.Arumugam ad K.A.Germa O Square sum graphs AKCE J.Graphs, Comb, 6(006) - 0 IJSER 03