Connective Eccentricity Index of Some Thorny Graphs

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1 Aals of ure ad Appled Matheatcs Vol. 7, No., 04, IN: X (), (ole) ublshed o 9 epteber 04 Aals of oectve Eccetrcty Idex of oe Thory raphs Nlaja De, k. Md. Abu Nayee ad Ata al 3 Departet of Basc ceces ad Huates (Matheatcs) alcutta Isttute of Egeerg ad Maageet, olkata, Ida Eal: de.laja@redffal.co Departet of Matheatcs, Alah Uversty, olkata, Ida Eal: ayee.ath@alah.ac. 3 Departet of Matheatcs, Natoal Isttute of Techology, Durgapur, Ida Eal: ata.bue@gal.co Receved 5 July 04; accepted August 04 Abstract. The coectve eccetrcty dex of a sple coected graph s defed as d ( v) ( ) =, where v V ( ) ε ( v) ε ( v ) ad respectvely deote the eccetrcty ad the degree of the vertex v. The thory graphs of are obtaed by attachg a uber of thors.e., degree oe vertces to each vertex of. I ths paper, we derve explct expressos for the coectve eccetrcty dex of soe classes of thory graphs. eywords: Eccetrcty; graph varat; coectve eccetrc dex; thor graph; oroa product. AM Matheatcs ubject lassfcato (00): 0535, 0507, Itroducto Let be a sple coected graph wth vertex set V ( ) ad edge set E( ). Also let ad deote the uber of vertces ad edges of the graph. The coectve eccetrcty dex of a graph was troduced by upta, gh ad Mada [] ad was defed as d ( v) ( ) =, where v V ( ) ε ( v) ε ( v ) ad respectvely deote the eccetrcty ad the degree of the vertex v. I [], horba coputed soe bouds of coectve eccetrcty dex ad explct expresso for ths dex for two fte classes of dedrers. Oe of the preset authors, [3], preseted soe bouds for ths coectve eccetrc dex ters of dfferet graph varats. horba ad Malekja [4], copute the eccetrc coectvty dex ad the coectve eccetrc dex of a fte faly of fullerees. Yu ad Feg [5], derved soe upper or lower bouds for the coectve eccetrc dex ad foud the axal ad the al values of coectve eccetrcty dex 59

2 Nlaja De, k. Md. Abu Nayee ad Ata al aog all -vertex graphs wth fxed uber of pedet vertces. The preset authors have also studed that dex o soe graph operatos [6]. Let be a gve graph wth vertex set{ v, v,..., v } ad{ p, p,..., p} be a set of o-egatve tegers. The, the thor graph of deoted by ( p, p,..., p ) s obtaed by attachg p pedat vertces to v for each. Ths oto of thor graph was troduced by uta [7] ad a uber of study o thor graphs for dfferet topologcal dces are ade by several researchers the recet past. Very recetly, De [8, 9] studed dfferet eccetrcty related topologcal dces o thor graphs. I ths paper, we derve explct expressos for the coectve eccetrcty dex of soe classes of thory graphs.. The Thory coplete graph Let be the copete graph wth vertces. The thory graph by attachg p thors at every vertex of thors attached to. Theore.. The coectve eccetrcty dex of s obtaed fro, =,,,. Let T be the total uber of 5 s gve by ( ) = E( ) + T. 6 roof: Let the vertces of are deoted by v, =,,...,, ad the ewly attached pedet vertces are deoted by v j, =,,..., ; j =,,..., p. Therefore, the degree ad eccetrcty of the vertces of are gve by = + p, =, ε ( v ) =, ε ( v ) = 3, for =,,..., ; j =,,..., p. j Thus the coectve eccetrcty dex of s gve by ( ) p j ε ( v ) ε ( v ) = = j= p + p = = j= 3 ( ) = + p + = 3 = fro where the desred result follows. j p ( ) 5 = + T 6 3. The Thory coplete bpartte graph Let, be the coplete bpartte graph wth (+) vertces. learly, the eccetrcty of the vertces of, are equal to two; ad there are uber of vertces of degree ad uber of vertces of degree. o the coectve eccetrcty dex of, s. Let j 60

3 oectve Eccetrcty Idex of oe Thory raphs, be the thory coplete bpartte graph obtaed fro, by attachg a uber of pedet vertces to each vertex of,. The we get the followg result. Theore 3.. The coectve eccetrcty dex of s gve by 7 ( ) = ( ) + T.,, roof: Let the vertex set of, s gve by { v, v,..., v, u, u,..., u} ad p, =,,..., ad p, =,,..., be the uber of pedet vertces attached to v ad u respectvely to obta,. Let, the ewly attached pedet vertces are deoted by v j, =,,..., ; j =,,..., p ad u j, =,,..., ; j =,,..., p. The the degree ad eccetrcty of the vertces of are gve by = + p, =,,,, ε ( v ) = 3, ε ( v ) = 4, for =,,..., ; j =,,..., p j ad d ( u ) = + p,,,, d ( u ) =, ε ( u ) = 3, ε ( u ) = 4, for =,,..., ; j =,,..., p. Thus, the j j,,, coectve eccetrcty dex of s gve by ( ),,, d ( u ) p ( ) p ( ) d v j d u j,,,, = ε ( v ) ε ( u ) ε ( v ) ε ( u ) = = = j= = j= j j,,,, p p + p + p = = = = j= 4 = j= 4 = + + p p p p = = 4 = = j fro where we get the desred result. 4. The Thory star 3 Let, =,( ) be the star graph o vertces. learly, ( ) = ( ). Let the thory graph obtaed by jog p, to each vertex of v, =,,. Theore 4.. The coectve eccetrcty dex of thor star s gve by 5 7 p ( ) = ( ) + T where p s the uber of pedet vertces added to the cetral vertex of. be roof: Let be the thory graph of obtaed by attachg p pedet vertces to each vertex v ( =,,..., ) of so that the degree ad eccetrcty of the vertces of 6

4 Nlaja De, k. Md. Abu Nayee ad Ata al are gve by = + p, = + p, =,...,, =, ε ( v ) =, j ε ( v ) = 3, =,...,, ε ( v ) = 4, for =,..., ; j =,,..., p, j ε ( v ) = 3 for j j =,,..., p. o the coectve eccetrcty dex of ( ) p j ε ( v ) ε ( v ) = = j= p p j j = ε ( v ) ε ( v ) ε ( v ) ε ( v ) = j= = j= j j + p + p p p = = 3 j= 3 = j= 4 j p p = + + p p 3 = = 5 p p = ( ) + p p = = fro where the desred result follows. 5. The Thory cycle Let be the cycle wth vertex set { } by attachg p, thors at every vertex of 6 s gve by v, v,..., v. The thory cycle, =,,. s obtaed fro Theore 5.. The coectve eccetrcty dex of the thory cycle s gve by 4 T ( + 3) 4 +, f s eve ( )( 4) ( ) ( ) = 4 T ( + ) 4 +, f s odd. ( + )( + 3) ( + ) roof: The degree ad eccetrcty of the vertces of are gve by = p +, ( ), + + d v = ε ( v ) =, f s odd ad ε ( v ) =, f s eve; j ε ( v ) =, f s odd ad ε ( v ) =, f s eve; for j j =,,..., ; j =,,..., p. o, whe s a odd uber, the coectve eccetrcty dex of s gve by ( ) p j ε ( v ) ε ( v ) = = j= j

5 oectve Eccetrcty Idex of oe Thory raphs p p + = ( + ) = j= ( + 3) p = ( p + ) + ( + ) = ( + 3) = j= T = ( T + ) + ( + ) ( + 3) fro where we get the desred result. roceedg slarly, f s a eve teger, the desred result follows. 6. Thory path Let deote a path graph o vertces. The thory graph of path graph s deoted by ad s obtaed by attachg a uber of degree oe vertces to every vertex of. the followg we fd coectve eccetrcty dex of Theore 6.. The coectve eccetrcty dex of s gve by. I p + p + 4 p + p p + p + + +, whe = + = = ( ) = p + p + 4 p + p p + p p + p , whe = +. = = roof: Whe the uber of vertces of s eve, say =+, let the vertces of are cosecutvely deoted by v, v,... v, v, v 0, v0, v, v,..., v, v where v 0 ad v0 are the ceters of the path + wth eccetrcty (+). We attach p ad p uber of pedet vertces to each v ad v respectvely (=,, ). The the degree ad eccetrcty of the other vertces of + s gve by = + p, = + p, d ( v ) = + p, d ( v ) = + p, = 0,,...,, = = d ( v ) for j j = 0,,..., ad j =,,..., p, ε ( v ) = + + = ε ( v ), for = 0,,...,, j ε ( v ) = = ε ( v ) for = 0,,..., ; j =,,..., p. j o the coectve eccetrcty dex of where, + s gve by ( ) = ( ) + ( ) d ( v ) ( ) d v ( ) = + ε ( v ) ε ( v ) = 0 0 = + p + p + p + p =

6 Also, Nlaja De, k. Md. Abu Nayee ad Ata al p + p + 4 p + p = p d ( v ) p ( ) j d v j ( ) = + ε ( v ) ε ( v ) = j= j j = = j p p = + = j= = j= p + p =. = obg, the desred result follows. Aga, f the uber of vertces of s odd, say =+, the a slar fasho we get the desred result. REFERENE..upta, M.gh ad A..Mada, oectve eccetrcty Idex: A ovel topologcal descrptor for predctg bologcal actvty, Joural of Molecular raphcs ad Modellg, 8 (000) M.horba, oectve eccetrc dex of fullerees, Joural of Matheatcal Naoscece, (0) N.De, Bouds for coectve eccetrc dex, Iteratoal Joural of oteporary Matheatcal ceces, 7(44) M.horba ad.malekja, A ew ethod for coputg the eccetrc coectvty dex of fullerees, erdca Joural o oputg, 6 (0) Yu ad L.Feg, O coectve eccetrcty dex of graphs, MATH oucatos Mathatcal ad oputer hestry, 69 (03) N.De, A.al ad.m.a.nayee, oectve eccetrc dex of soe graph operatos, arxv: I.uta, Dstace thory graph, ublcatos de l'isttut Mathéatqu, 63 (998) N.De, O eccetrc coectvty dex ad polyoal of thor graph, Appled Matheatcs, 3 (0) N.De, Augeted eccetrc coectvty dex of soe thor graphs, Iteratoal Joural of Appled Matheatcs Research, (4) (0)