Vertex Odd Divisor Cordial Labeling of Graphs

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1 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 0, October 0. Vertex Odd Dvsor Cordal Labelg of Graphs ISSN A. Muthaya ad P. Pugaleth Assstat Professor, P.G. ad Research Departmet of Mathematcs, Govt. Arts College, Aryalur 6,Taml Nadu, Ida. Research Scholar, P.G. ad Research Departmet of Mathematcs, Govt. Arts College, Aryalur 6,Taml Nadu, Ida. Abstract I ths paper, the vertex odd dvsor cordal labelg of wheel graph W, swtchg of a pedet vertex path P, swtchg of a vertex cycle C, Bstar B, S( ), B, DS(B ), S (B ) ad < (), preseted., () (), > are eywords: Dvsor Cordal labelg, Dvsor Cordal Graphs, Vertex Odd Dvsor Cordal labelg, Vertex Odd Square Dvsor Cordal Graphs.. Itroducto By a graph, we mea a fte, udrected graph wthout loops ad multple edges, for terms ot defed here, we refer to Harary [4]. For stadard termology ad otatos related to umber theory we refer to Burto [] ad graph labelg, we refer to Galla []. I [], Caht troduce the cocept of cordal labelg of graph. I [], Varatharaja et al. troduce the cocept of dvsor cordal labelg of graph. The dvsor cordal labelg of varous types of graph are preseted [-8, 0-]. I [], Muthaya et al troduce vertex odd dvsor cordal labelg of graph. Further they proved path, cycle,,,,m, helm H, flower Fl, <, (), () >, swtchg of the apex vertex helm H ad S ( ) are vertex odd dvsor cordal graphs uder some codtos. Every dvsor cordal graphs eed ot be vertex odd dvsor cordal graphs. Also, every vertex odd dvsor cordal graphs eed ot be dvsor cordal graphs. The bref summares of defto whch are ecessary for the preset vestgato are provded below.. Deftos Defto :. A graph labelg s the assgmet of uque detfers to the edges ad vertces of a graph. Defto :. A mappg f :V(G) {0,} s called bary vertex labelg of G ad f(v) s called the label of the vertex v of G uder f. If for a edge e = uv, the duced edge labelg f* : E(G) {0,} s gve by f*(e) = f(u) f(v). The v f () = umber of vertces of havg label uder f ad e f () = umber of edges of havg label uder f*. A bary vertex labelg f of a graph G s called a cordal labelg f v f (0) v f () ad e f (0) e f (). A graph G s cordal f t admts cordal labelg. Defto :. Let a ad b be two tegers. If a dvdes b meas that there s a postve teger k such that b = ka. It s deoted by a b. If a does ot dvde b, the we deote a b. Defto :.4 Let G = (V(G), E(G)) be a smple graph ad f : V(G) {,,..., V(G) } be a bjecto. For each edge uv, assg the label f f(u) f(v) or f(v) f(u) ad the label 0 otherwse. The fucto f s called a dvsor cordal labelg f e f (0) e f (). A graph wth a dvsor cordal labelg s called a dvsor cordal graph. Defto :. Let G = (V(G), E(G)) be a smple graph wth vertces ad f : V {,,, } be a bjecto. For each edge uv, assg the label f ether f(u) f(v) or f(v) f(u) ad the label 0 otherwse. f s called a vertex odd dvsor cordal labelg f e f (0) e f (). A graph wth vertex odd dvsor cordal labelg s called a vertex odd dvsor cordal graph. Defto :.6 The jo G = G +G of graphs G ad G wth dsjot vertex sets V ad V, ad edge sets E ad E, s the graph uo G G together wth all the edges jog

2 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 0, October 0. V ad V. A wheel graph W s defed as C +, where C deotes the cycle wth vertces. ISSN vertex x m where m t ad G has t(+) vertces ad t(+) edges. Defto :. For a graph G the splttg graph S (G) of a graph G s obtaed by addg a ew vertex v correspodg to each vertex v of G such that N(v) = N(v ). Defto :.8 Let G = (V(G), E(G)) be a graph wth vertex set V = S S S T where each S s a set of vertces havg at least two vertces of the same degree ad T = V\ S. The degree splttg graph of G deoted by DS(G) s obtaed from G by addg vertces w,w,w,, w t ad jog to each vertex of S for t. Defto :. For a smple coected graph G the square of graph G s deoted by G ad defed as the graph wth the same vertex set as of G ad two vertces are adjacet G f they are at a dstace or apart G. Defto :.0 Bstar B s the graph obtaed by jog the ceter (apex) vertces of two copes of by a edge. Defto :. A subdvso of the edge e = uv of a graph G s the replacemet of the edge e by a ew vertex w ad two ew edges uw ad wv. Ths operato s also called a elemetary subdvso of G. A graph H obtaed by a sequece of elemetary subdvsos from a graph G s sad to be a subdvso graph of G. It s deoted by S(G). Defto :. A vertex swtchg G v of a graph G s obtaed by takg a vertex v of G, removg the etre edges cdet wth v ad addg edges jog v to every vertex whch are ot adjacet to v G. Defto :. Cosder t copes of stars amely,,..., The G = <, (), (),..., jog apex vertces of each () () (t). (t) > s the graph obtaed by (m ) ad (m) to a ew. Ma Results Theorem. The wheel graph W = C + s vertex odd dvsor cordal graph, where. Let G be a wheel graph W = C +. Let v be the cetral vertex ad v,... be the vertces of C. The V(G ) = + ad E(G) =. Defe vertex labelg f : V(G) {,,,,+} as f(v) =, Case : (mod ) f(v ) = +,. f(v ) = +, f(v ) =. The e f () = e f (0) =. Case : 0, (mod ) f(v ) = +,. The e f () = e f (0) =. I both the cases, we have e f (0) e f (). Therefore, W s vertex odd dvsor cordal graph. Example. The vertex odd dvsor cordal labelg of W 4 s show fgure.. Theorem. Fgure. Swtchg of a vertex cycle C admts vertex odd dvsor cordal labelg. Let v, be the successve vertces of C. G v deotes graph s obtaed by swtchg of vertex v of G = C. Wthout loss of geeralty let the swtched vertex be v. The V( G ) = ad E( G ) =. v v 6

3 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 0, October 0. Defe vertex labelg f : V( G v ) {,,,, } as f(v ) =, f(v ) = for The, e f () = ad e f (0) =. Thus, e f (0) e f (). Hece, the graph obtaed by swtchg of a vertex cycle C s a vertex odd dvsor cordal graph. Example. The vertex odd dvsor cordal labelg of swtchg of a vertex cycle C 8 s show fgure.. Theorem. Fgure. Swtchg of a pedet vertex path P s vertex odd dvsor cordal graph. Let v,, be the vertces of path P. The graph G s obtaed by swtchg of a pedet vertex path P. v ad v are pedet vertex of path P. Wthout loss of geeralty, let the swtched vertex be v. The V(G ) = ad E(G) = 4. Defe vertex labelg f : V(G) {,,,, } as f(v ) =,. The e f (0) = e f () =. Therefore, e f (0) e f (). Hece, G s vertex odd dvsor cordal graph. Example. The vertex odd dvsor cordal labelg of swtchg of a pedet vertex path P 6 s show fgure.. Theorem.4 Fgure. ISSN The Bstar B s vertex odd dvsor cordal graph, where. Let B be a graph wth vertex set {u,v,u,v, }, where u are pedat vertces. Let G be the graph B. The vertex set V(G) = {u, w, u : } ad the edge set E(G) = {uw, uu v, : }. The V(G) = + ad E(G) = +. p s the largest prme umber such that p 4+. Defe vertex labelg f : V(G) {,,...,4+} as f(u) =, f(v) = p ad assg the remag labels to the remag vertces v,v,,v,u,u,,u. The e f (0) = ad e f () = +. Therefore, e f (0) e f (). Hece G s vertex odd dvsor cordal graph. Example.4 The vertex odd dvsor cordal labelg of B 4,4 s show fgure.4. Theorem. Fgure.4 The graph S( ) s a vertex odd dvsor cordal graph. Proof Let G be a subdvso of the star. Let V(G) = {v, u : } ad E(G) = {vv,v u : }. The G = S( ). Defe vertex labelg f : V(G) {,,,,4+} as follows f(v) =, f(v ) = + 4( ) for, f(u ) = + 4( ) for.

4 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 0, October 0. The, e f (0) = e f () =. Therefore e f (0) e f (). Hece S( ) s a vertex odd dvsor cordal graph. Example. The vertex odd dvsor cordal labelg of S(, ) s show fgure.. Fgure. Theorem.6 B s a vertex odd dvsor cordal graph. Let B be a graph wth vertex set {u,v,u,v, }, where u are pedat vertces. Let G be the graph B The V(G) = + ad E(G) = 4+. Let p be the largest prme umber p 4+. Defe vertex labelg f : V(G) {,,,, 4+} as f(u) =, f(v) = p, Assg the remag labels to the remag vertces u, u,, u,. The, e f (0) =, e f () = +. Thus, e f (0) e f (). Hece, B s a vertex odd dvsor cordal graph. Example.6 Theorem. Fgure.6 ISSN S (B ) s a vertex odd dvsor cordal graph. Let B be a graph wth vertex set {u,v,u,v, }, where u are pedat vertces. I order to obta S (B ), add u, u vertces correspodg to u, u, where. Let G be a graph S (B ). The V(G) = 4(+) ad E(G) = 6 +. Let p ad p be the largest ad ext largest prme umbers, such that p < p 8+. Defe vertex labelg f :V(G) {,,,, 8+} as f(u) =, f(u ) =, f(v) = p, f(v ) = p, f(u ) = + 6( ) for f( u ) = f(u ) for ) = f(u ) 4 for Assg the remag labels to the remag vertces v,. The, e f (0) = + ad e f () = +. Thus, e f (0) e f (). Hece, S (B ) s a vertex odd dvsor cordal graph. Example. The vertex odd dvsor cordal labelg of S (B, ) s show fgure.. The vertex odd dvsor cordal labelg of B, show fgure.6. s Fgure. 8

5 IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 0, October 0. Theorem.8 Theorem. ISSN DS(B ) s a vertex odd dvsor cordal graph. Let B be a graph wth vertex set {u,v,u,v, }, where u are pedat vertces. Here, V(B ) = V V, where V = {u : } ad V = {u}. I order to obta DS(B ) from B, add w, w correspodg to V ad V. Let G be a graph DS(B ). The, V(G) = +4 ad E(G) = 4+. Let p be the largest prme umber such that p 8+. Defe vertex labelg f : V(G) {,,,, 8+} as f(u) =, f(v) = p, f(w ) =, f(w ) =, Case : s eve ad k = / f(u ) = + 6( ) for k f(u ) = + 6( k ) for k+. The, e f (0) = + ad e f () = +. Case : s odd ad k = (+)/ f(u ) = + 6( ) for k f(u ) = + 6( k ) for k+. The, e f (0) = + ad e f () = +. I both cases, e f (0) e f (). Hece, DS(B ) s a vertex odd dvsor cordal graph. Example.8 The vertex odd dvsor cordal labelg of DS(B, ) s show fgure.8. Fgure.8 The graph G = < () () (),,, > s vertex odd dvsor cordal. Let G = < () () (),,, > be a graph wth + vertces ad + 4 edges. () () () Let v,... be the pedat vertces of (), () ad let c be the apex vertex of for =,,. The V(G) = + ad E(G) = +4. Here, c ad c are adjacet to x ad c ad c are adjacet to x. Let p be the largest prme umber such that p 6+. Defe vertex labelg f : V(G) {,,,, 6+ } as f(c ) = p, f(c ) =, f(c ) =, f(x ) =, f(x ) =, Case : s eve ad k = / () ) = + 6( ) for k () ) = + 6( k ) for k+. Assg the remag labels to the remag vertces () () () () () () v,..., The, e f (0) = e f () =. Case : s odd ad k = (+)/ () ) = + 6( ) for k () ) = + 6( k ) for k+. Assg the remag labels to the remag vertces () () () () () () v,..., The, e f (0) = ad e f () =. I both cases, e f (0) e f (). Hece, the graph G = < () () (),,, > s vertex odd dvsor cordal graph. Example. The vertex odd dvsor cordal labelg of the graph, > s show fgure.. < (),, (), (),

6 4. Coclusos IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Issue 0, October 0. Fgure. I ths paper, the vertex odd dvsor cordal labelg of wheel graph W, swtchg of a pedet vertex path P, swtchg of a vertex cycle C, Bstar B, S( ), B, DS(B ), S (B ) ad < proved. (), (), () > are ISSN Appled Mathematcs, Vol 4, No., 0, pp. 0-. [] R. Varatharaja, S. Navaaeethakrsha ad. Nagaraja, Dvsor cordal graphs, Iteratoal J. Math. Comb., Vol 4, 0, pp. -. [] R. Varatharaja, S. Navaaeethakrsha ad. Nagaraja, Specal classes of dvsor cordal graphs, Iteratoal Mathematcal Forum, Vol, No., 0, pp. -4. Refereces [] I. Caht, Cordal graphs: A weaker verso of graceful ad harmoous graphs, Ars Combatora, Vol, 8, pp [] Davd M. Burto, Elemetary Number Theory, Secod Edto, Wm. C. Brow Compay Publshers, 80. [] J. A. Galla, A dyamc survey of graph labelg, The Electroc Joural of Combatorcs, 6, # DS6, 0. [4] F. Harary, Graph Theory, Addso-Wesley, Readg, Mass,. [] P. Lawrece Rozaro Raj ad R. Vall, Some ew famles of dvsor cordal graphs, Iteratoal Joural of Mathematcs Treds ad Techology, Vol, No., 04. [6] P. Lawrece Rozaro Raj ad R. Lawrece Joseph Maohara, Dvsor Cordal Labellg of Some Dscoected Graphs, Iteratoal Joural of Mathematcs Treds ad Techology, Vol., No., 04, pp 4-6. [] P.Maya ad T.Ncholas, Some New Famles of Dvsor Cordal Graph, Aals of Pure ad Appled Mathematcs Vol., No., 04, pp. -4. [8] A. Muthaya ad P. Pugaleth, Some ew dvsor cordal graphs, Iteratoal Joural of Mathematcs Treds ad Techology, Vol, No., 04. [] A. Muthaya ad P. Pugaleth, Vertex odd dvsor cordal graphs, commucated. [0] S.. Vadya ad N. H. Shah, Some Star ad Bstar Related Dvsor Cordal Graphs, Aals of Pure ad Appled Mathematcs, Vol, No., 0, pp. 6-. [] S.. Vadya ad N. H. Shah, Further Results o Dvsor Cordal Labelg, Aals of Pure ad 400