Australian Journal of Basic and Applied Sciences, 5(11): , 2011 ISSN On tvs of Subdivision of Star S n

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1 Australia Joural of Basic ad Applied Scieces 5(11): ISSN O tvs of Subdivisio of Star S 1 Muhaad Kara Siddiqui ad Deeba Afzal 1 Abdus Sala School of Matheatical Scieces G.C. Uiversity Lahore Pakista. Abstract: A vertex irregular total t -labelig : V E {1 t} of a graph G =( V E ) is a labelig of vertices ad edges of G i such a way that for ay differet vertex u ad v their total weights are distict. The total vertex irregularity stregth tvs( G ) is defied as the iiu t for which G has a vertex irregular total t - labelig. I this paper we deteried the exact value of the total vertex irregularity stregth of subdivisio of star S. Key words: Vertex irregular total t -labelig irregular assiget total vertex irregularity stregth subdivisio of star. INTRODUCTION Chartrad Jacobso Lehel Oellera Ruiz ad Saba (1988) itroduced labeligs of the edges of a graph G with positive itegers such that the su of the labels of edges icidet with a vertex is differet for all the vertices. Such labeligs were called irregular assigets ad the irregularity stregth sg ( ) of a graph G is kow as the iiu k for which G has a irregular assiget usig labels at ost k. The irregularity stregth sg ( ) ca be iterpreted as the sallest iteger k for which G ca be tured ito a ultigraph G by replacig each edge by a set of at ost k parallel edges such that the degrees of the vertices i G are all differet. Fidig the irregularity stregth of a graph sees to be hard eve for graphs with siple structure see (Boha T. ad D. Kravitz 00). Karo ski Luczak ad Thoaso (00) cojectured that the edges of every coected graph of order at least ca be assiged labels fro {1} such that for all pairs of adjacet vertices the sus of the labels of the icidet edges are differet. Motivated by irregular assigets Ba c a Jedro l Miller ad Rya (007) defied a vertex irregular total k -labelig of a ( pq ) -graph G =( V E ) to be a labelig of the vertices ad edges of G : V E {1 k} such that the total vertex-weight wt( x)= ( x) ( xy) are differet for all xye vertices that is wt( x) wt( y) for all differet vertices x y V. Furtherore they defied the total vertex irregularity stregth tvs( G ) of G as the iiu k for which G has a vertex irregular total k -labelig. It is easy to see that irregularity stregth sg ( ) of a graph G is defied oly for graphs cotaiig at ost oe isolated vertex ad o coected copoet of order. O the other had the total vertex irregularity stregth tvs( G ) is defied for every graph G. If a edge labelig f : E {1 s( G)} provides the irregularity stregth sg ( ) the we exted this labelig to total labelig i such a way. ( xy)= f ( xy) for every xy E( G) ( x)=1 for every x V ( G). Thus the total labelig is a vertex irregular total labelig ad for graphs with o copoet of order is Correspodig Author: Muhaad Kara Siddiqui Abdus Sala School of Matheatical Scieces G.C. Uiversity Lahore Pakista. E-ail: karasiddiqui75@gail.co 16

2 Aust. J. Basic & Appl. Sci. 5(11): tvsg ( ) sg ( ). Nierhoff (000) proved that for all ( pq ) -graphs G with o copoet of order at ost ad G K the irregularity stregth sg ( ) p 1. Fro this result it follows that tvs( G) p 1. (1) I (Ba c a M. et al. 007) several bouds ad exact values of tvs( G ) were deteried for differet types of graphs (i particular for stars cliques ad priss). Aog others the authors proved that for every ( pq ) -graph G with iiu degree = ( G) ad axiu degree = ( G) p ( G) tvs( G) p ( G) ( G) 1. ( G) 1 () I the case of r -regular graphs () gives p r tvs( G) p r 1. r 1 () For graphs with o copoet of order Ba c a et al. i (007) stregtheed also these upper bouds p provig that tvs( G) p 1 ( G) 1. These results were the iproved by Przybylo i (009) for sparse graphs ad for graphs with large iiu degree. I the latter case were proved the bouds p p tvs( G)< 8 i geeral ad tvs( G)<8 for r -regular ( pq ) -graphs. Aholcer Kalkowski ( G) r ad Przybylo [] established a ew upper boud of the for p tvs( G) 1. ( G) () Wijaya Slai Surahat ad Jedro l (005) deteried a exact value of the total vertex irregularity stregth for coplete bipartite graphs. I Ahad ad Ba c a foud exact values of tvs for Jahagir graphs ad circulat graphs. Wijaya ad Slai (008) foud the exact values of tvs for wheels fas sus ad friedship graphs. I Ahad (011) foud the exact value of tvs of covex polytope graphs. I Nurdi Baskoro Sala ad Goas (010) deteried exact values of tvs for several types of trees ad for disjoit uio of paths i (Nurdi E.T. et al.) ad (Nurdi E.T. et al. 009) respectively. Moreover Nurdi Baskoro Sala Goas (010) proved the followig lower boud of tvs for ay graph G. Theore 1: (Nurdi E.T. Baskoro et al. 010) Let G be a coected graph havig i vertices of degree i ( i = 1 ) where ad are the iiu ad the axiu degree of G respectively. The i 1 i= tvs( G) ax. 1 (5) 1 The ai purpose of this paper is deteried the exact values of the total vertex irregularity stregth of subdivisio of star S. 17

3 Aust. J. Basic & Appl. Sci. 5(11): Subdivisio of Star S : I (Sala A.N.M. et al. 010) for 0 ad to every edge of a star S. Thus the star S ca be writte as let S be a graph obtaied by isertig vertices S 0. Fig. 1: The graph of S. We defie the vertex-set ad the edge-set of the graph S as follows: V( S )={ c x : i[1 ] j[1 1]} E( S )= A A i j i1 i j Where: A ={ cx : i [1 ]} i1 i1 Clearly a graph i j i j 1 i j A = { x x : i[1 ] j[ 1]}. S has 1 vertices ad edges. Aog these vertices oe vertex has degree vertices have degree oe ad the reaiig vertices have degree two. The ext theores ad leas deteries the exact value of the total vertex irregularity stregth of subdivisio of star. 1 1 Theore : Let. The tvs( S 1 Proof: Sice the graph S has 1 vertices aog these vertices oe vertex has degree vertices have 1 1 degree 1 ad the reaiig vertices have degree. Therefore fro (5) we have tvs( S). 18

4 Aust. J. Basic & Appl. Sci. 5(11): Put t =. To show that t is a upper boud for 1 tvs( S ) we describe a total t -labelig : V E {1 t} as follows: for all 1 i ()= c t if j 1 ( xij ) i if j i 1 t 1 if i 1( od) i 1) t1 if i ( od) i t 1 if i 0( od) i 1 i 1( ) f i od i ) ifi ( od) i 1 if i 0( od) Uder the labelig the total vertex-weights are described as follows: i i1 i 1( ) f i od i i wt( xi 1) 1 if i ( od) 1 i if i 0( od) i i 1 1( ) if i od i wt( xi) if i ( od) 1 i if i 0( od) wt()= c ( t ) S S S 1 19

5 Aust. J. Basic & Appl. Sci. 5(11): Where: i1 i i S = S = S = 1 i1( od ) i( od ) i( od ) It is easy to verify that all vertex ad edge labels are at ost t ad the total vertex-weights are differet for all 1 pairs of distict vertices. Thus the labelig provide the upper boud o tvs( S ). Cobiig with lower boud 1 1 we coclude that tvs( S 1 Theore : Let. The tvs( S Proof: Sice the graph 1 S has 1 vertices. Therefore fro (5) we have tvs( S ). 1 Put t =. To show that t is a upper boud for tvs( S ) we describe a total t -labelig : V E {1 t} with ()=1 c as follows: t if 1ia d j =1 ( xi j)= i 1 if 1 i t 1a d j = i if 1ia d j = i1 if 1 it1 a d j = 1 j)= 1 if 1 i a d j = 1 if 1 i a d j = Uder the labelig the total vertex-weights are described as follows: ti if 1it1a d j =1 wt( xi j)= 1 i if 1 i t 1a d j = 1 i if 1 i a d j = tt ( 1) wt()= c It is easy to see that the total vertex-weights are differet for all pairs of distict vertices. I fact 1 tvs( S ). Cobiig with lower boud we coclude that 1 tvs( S Lea 1: tvs( S )=5. Proof: Sice 1 vs ( ) =1 the fro (5) it follows that : V E {1 5} proves the coverse iequality. tvs( S ) 5. The existece of the optial labelig 150

6 Aust. J. Basic & Appl. Sci. 5(11): ()= c 1( x1)= 1( x)= 1 x1 1 x 1 x ( )=1 ( )= ( )= ( )= ( )= ( )= ( )=1 ( )= 1 x1 1 x 1 x 1 xi xi 1 xi xi ( x )= ( x )= ( x )= ( x )=5 for all 1 i ( x x )= ( cx )= i. for all 1 i 1 i1 i 1 i1 Thus tvs( S ) 5. Cobiig with the lower boud we get tvs( S )=5. Theore : Let. The 1 tvs( S Proof: Sice the graph 1 S has 1 vertices therefore fro (5) we have tvs( S ). Put 1 t =. To show that t is a upper boud for tvs( S ) we describe a total t -labelig : V E {1 t} for 1 i as follows: ()=1 c t if j =1 i if j = ( xij )= i if j = i if j = i1 t1 1( ) if i od i 1) = t1 if i ( od) i t1 if i 0( od) i 1 1( ) if i od i )= if i ( od) i 1 if i 0( od) )= if 1i 151

7 Aust. J. Basic & Appl. Sci. 5(11): i 1 1( ) if i od i )= if i ( od) i 1 if i 0( od) Uder the labelig the total vertex-weights are described as follows: i i1 1( ) if i od i i wt( xi1)= 1 if i ( od) 1 i if i 0( od) Now for all j i i ( j ) 1 1( ) ifi od i wt( xi j) = ( j) if i ( od) ( j ) 1 i ifi0( od) wt()=1 c ( t 1) S S S 1 where S1 S S are sae as i Theore. It is routie atter to check that all vertex ad edge labels are at ost t ad the total vertex-weights are distict. 1 1 I fact tvs( S ). Cobiig with the lower boud we coclude that tvs( S Lea : tvs( S )=6. Proof: The existece of the optial labelig : V E {1 6} proves the required result. ( x )=1 ( x )=5 15 ( )= ( )= ( )= ( x )= ( x )= ( x )= ( )= ( )= ( )= ( )=6 x x x c x11 x1 x1 ( x )= ( x )= ( x )= ( x )=

8 Aust. J. Basic & Appl. Sci. 5(11): For all 1i ( )= ( )=1 ( )= ( xi xi)= ( xi1 xi)=6. cxi1 i xi xi5 xi xi 5 1 Theore 5: Let. The tvs( S 5 1 Proof: Sice the graph S has 5 1 vertices the fro (5) we havetvs( S ). 5 1 Put t =. To show that t is a upper boud for tvs( S ) we describe a total t -labelig : V E {1 t} 5 for 1i with ()= c t 5 as follows: t if j =1 t1 i if j = 5( xij )= i if j = i 1 if j = i if j =5 t1 i if j =1 t if j = 5 j)= if j = 1 if j = 1 if j = 5 Uder the labelig 5 the total vertex-weights for all 1 j 5 we have: wt( x )=(5 j) 1 i for 1i i j ( 1) wt()= c t ( t 1) It is routie atter to check that all vertex ad edge labels are at ost t ad the total vertex-weights are differet 5 1 for all pairs of distict vertices. I fact tvs( S ). Cobiig with the lower boud we coclude that 5 1 tvs( S Lea : tvs( S )=7. 5 Proof: The existece of the optial labelig proves the required result. ( )= ( )=5 ( )= ( )= ( )= x x x1 x x5 15

9 Aust. J. Basic & Appl. Sci. 5(11): ()= ( )= ( )= ( )= c x1 x5 x6 i j ( x )=1 ( x )=6 ( x )= ( x )= For all 1i ( x )=7 for i =1 ad j =1. ( cx )=5 ( x x )=1 i1 i5 i6 ( )= ( )= ( x x )= ( x x )= i. xi xi5 xi xi i i i1 i It is easy to see that total vertex-weights are distict therefore tvs( S )=7. 5 ( 1) 1 Theore 6: Let 5 8 ad. The tvs( S 1 Proof: Sice the graph S has 1 vertices therefore fro (5) we havetvs( S ). Put 1 t =. To show that t is a upper boud for tvs( S ) we describe a total t -labelig 6 for 1i as follows: ()= c t 6 t if j =1 t i if j =ad odd 6( xij )= ( 1) 1 t i if j =ad eve 1 i j if j 1 1 t i if j =1 t if j = 6 j)= if j = 1 j if j = 1 1 if j = 1 whe =6 ( A )= 1 6 i whe =7 if j = 6 j)= 1 if j = whe =8 15

10 Aust. J. Basic & Appl. Sci. 5(11): if j = 6 j)= if j = 1 if j =5 Uder the labelig 6 the total vertex-weights for 1 j 1 wt( x )=( 1 j) 1 i i j ( 1) wt()= c t ( t 1) ad 1 i we have: It is routie atter to check that all vertex ad edge labels are at ost t ad the total vertex-weights are differet 1 for all pairs of distict vertices. I fact tvs( S ). Cobiig with the lower boud we coclude 1 that tvs( S The results fro above leas ad Theores adds further support to a recet cojecture. Cojecture: (Nurdi E.T. Baskoro et al. 010) Let G be a coected graph havig i vertices of degree i ( i = 1 ) where ad are the iiu ad the axiu degree of G respectively. The tvs G 1 1 i 1 i= ( )= ax. Coclusio: I this paper we deterie the exact value of total vertex irregularity stregth of subdivisio of star for ad. We are tried to fid a irregular total without success. So we coclude the paper with the followig ope proble. -labelig for 9 but so far Ope Proble: For 9 deterie the total vertex irregular stregth of subdivisio of star S. REFERENCES Ahad A. ad M. Ba c a. O vertex irregular total labeligs Ars Cobi. i press. Ahad A tvs of covex polytope graphs with pedet edges Sci. it. (): Aholcer M. M. Kalkowski ad J. Przybylo 009. A ew upper boud for the total vertex irregularity stregth of graphs Discrete Math. 09: Ba c a M. S. Jedro l M. Miller ad J. Rya 007. O irregular total labelligs Discrete Math. 07: Boha T. ad D. Kravitz 00. O the irregularity stregth of trees J. Graph Theory 5: 1-5. Chartrad G. M.S. Jacobso J. Lehel O.R. Oellera S. Ruiz ad F. Saba Irregular etworks Cogr. Nuer. 6: Karo ski M. T. Luczak ad A. Thoaso 00. Edge weights ad vertex colours J. Cobi. Theory B. 91:

11 Aust. J. Basic & Appl. Sci. 5(11): Nierhoff T A tight boud o the irregularity stregth of graphs SIAM J. Discrete Math. 1: 1-. Nurdi E.T. Baskoro A.N.M. Sala ad N.N. Goas 009. O the total vertex irregularity stregth of a disjoit uio of t copies of path J. Cobi. Math. Cobi. Coput. 71: Nurdi E.T. Baskoro A.N.M. Sala ad N.N. Goas 010. O the total vertex irregularity stregth of trees Discrete Math. 10: Nurdi E.T. Baskoro A.N.M. Sala ad N.N. Goas. O total vertex irregular labeligs for several types of trees Utilitas Math. to appear. Przybylo J Liear boud o the irregularity stregth ad the total vertex irregularity stregth of graphs SIAM J. Discrete Math. : Sala A.N.M. A.A. Gede Ngurah ad N. Izzati 010. O (super) edge-agic total labeligs of a subdivisio of a star S Utilitas Matheatica. 81: Wijaya K. ad Slai 008. Total vertex irregular labelig of wheels fas sus ad friedship graphs JCMCC. 65: Wijaya K. Slai Surahat ad S. Jedrol 005. Total vertex irregular labelig of coplete bipartite graphs JCMCC. 55: