Odd-elegant Labeling Algorithm of Generalized Ring Core Networks

Transcription

1 6th Iteratoal Coferece o Machery Materals Eroet Botechology ad Coputer (MMEBC 06) Odd-elegat Labelg Algorth of Geeralzed Rg Core Networs Ja Xe a Bg Yao b ad Wee Hogc College of Matheatcs ad Statstcs Northwest Noral Uersty Lazhou Cha a College of Matheatcs Lazhou Cty Uersty Lazhou Cha College of Early Chldhood Teachers Lazhou Cty Uersty Lazhou Cha Eal:xj760@6co b Eal:yybb98@6co c Eal:hogw007@6co Keywords: geeralzed rg core etwor; odd- elegat feature; odd- elegat labelg; odd- elegat labelg algorth; effecteess of algorth Abstract I the etwor desg the choce of the etwor topology plays a decse role for the realzato of the fucto of coputer etwor effcecy The labelg proble of the coputer etwor topology drectly affects the etwor desg ad coucato costs etc Geeralzed rg etwor topology s a ery portat hybrd etwor topology structure ad geeralzed rg core etwor s ts base I ths paper based o the requreets of research of geeralzed rg etwor addressg the author desgs the GRN-OEL-algorth whe 0(od 4) proes odd-elegat of the geeralzed rg core etwor wors out the correspodg software ad tests the practcal effecteess of ths algorth wth our experetal data Itroducto Coputer etwor s the result of the close cobato utual peetrato utual prooto ad coo deelopet betwee coputer techology ad coucato techology I the etwor desg the choce of the etwor topology plays a decse role for the realzato of the fucto of coputer etwor effcecy The labelg proble of the coputer etwor topology drectly affects the etwor desg ad coucato costs etc The labelg of etwor topology refers to a appg of teger set to the ode or edge of etwor topology ad t satsfes certa codtos Accordg to dfferet codtos the researchers defed seeral types of labelg of etwor topology ad put forward ay cojectures I 98 Chag put forward the cocept of elegat labelg ad guessed that []: all tree topology are elegat I 0 Zhou et al defed the cocept of odd-elegat labelg ad put forward the hypothess that []: all tree topology are odd-elegat Bus topology star topology ad rg etwor topology are the three basc etwor topologes ad the labelg probles of the lays a ecessary theoretcal bass for the desg of the coputer etwor syste Howeer the sgleess of ther structure serously affects the exteso of etwor fucto Therefore the ew topology structure has bee orgac cobed wth two or ore a sgle topology structure whch has becoe a portat research topc for researchers of coputer theory ad applcato especally for the web worer[-6] I 007 Gao Zheb proed the odd-graceful of three uo structure P S ad C ( s a ultple of 4) [7] I 009 Barretos proed the odd-graceful of topology structure of tree wth ts daeter of o ore tha fe [8] I 0 Zhou et al proed the odd-elegat of lobster -----a hybrd topology structure [] Geeralzed rg etwor s a ery portat hybrd etwor topology structure whch refers to a uber of closed rgs fored of odes led together pot to pot ad ed to ed etwor ad each odes of rgs has a le odes(we call le odes leaes) The forato both ca be traslated a drecto betwee odes the loop ad ca be traslated betwee rg odes ad leaes After the leaes of geeralzed rg etwor are reoed we call the rests the geeralzed rg core etwor Obously geeralzed rg core etwor s the bass of geeralzed rg etwor I ths paper based o the requreets of research of geeralzed rg etwor addressg the author desgs the GRN-OEL algorth whe 0(od 4) proes odd-elegat of ths 06 The authors - Publshed by Atlats Press 86

2 etwor topology wors out the correspodg software ad tests the practcal effecteess of ths algorth wth our experetal data Prelary owledge We beg wth sple fte ad udrected etwor topology G = ( V E) wth V for odes set ad E for edges set For the etwor topology G poste teger p s called as the odes uber of G ad q s called as the edges uber of G For the sae of splcty the shorthad sybol [ ] stads for a teger set { + } where ad are tegers wth 0 < Defto [] A fucto f s called odd-elegat labelg of a graph G f f : V [0 q ] s jecte ad the duced fucto f*: EG ( ) { q } defed as f *( e) = f( u) + f( ) od q ( e = u) s bjecto The graph whch adts odd-elegat labelg s called a odd-elegat graph Defto Let s a teger ot less tha ad let 0(od 4) are all poste teger The odes each rg C ( [ ]) wll be ordered to the sae drecto Ad let the ode ( ) of of rg C ( [ ]) s cocded wth frst ode ( ) of rg C the a etwor topology G = ω cludg rgs s defed whch s called as geeralzed rg core etwor f 0(od 4) (fgure ) Odd-elegat labelg algorth of geeralzed rg core etwor Let G = ω be a geeralzed rg core etwor f N ad 0 (od 4) the: V = { ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ad p = + = q = = } The ode bee oerlap labelg s regarded as two dfferet odes ( ) ad ( [ ] ) All odes ca be dded to ( ) three parts: 4 /- / /+ / /+ 4 /+ / / /- / - /+ / /- / - /+ /+ Fg Geeralzed rg core etwor wth labeled ertces { ( ) [ ] [ V = j j ]} V 4 = { ( j ) [ ] j [ + ]} V 4 = { ( ) [ ] [ j j ]} Edge set E = { j( j+ ) [ ] j [ ]} accordg to the locato of rg ca be dded to parts: E= { ( ) } E { ( ) } E = { } ( ) = O the bass of the aboe classfcato we costruct a odd-elegat labelg algorth of geeralzed rg core etwor ω (here referred to as GRN-OEL- ALGORITHM ) ad t s as follows (Table ) 86

3 Table GRN-OEL-ALGORITHM Algorth: GRN-OEL- ALGORITHM Iput: The uber of rgs of geeralzed rg core etwor ω The uber of odes ( = ) of each rg C Output: Odd-elegat labelg of geeralzed rg core etwor ω Costruct a geeralzed rg core etwor ω wth p odes ad q edges ( p = + let + = 0 = q = ) ad Label the ode ( ) ( [ ] [ j j ]) accordg to the labelg fucto f( ) = ( ) 4 ( j ) + j Label the ode ( j ) ( [ ] j [ + ]) accordg to the labelg fucto f( ) = 4 ( j ) + j 4 Label the ode ( )( [ ] [ j j ]) accordg to the labelg fucto f( ) = ( j) + j 5 Label the edge u E of ω accordg to the fucto g( u) = f ( u) + f ( ) od q for u VG ( ) u 6 Output a odd-elegat labelg of geeralzed rg core etwor ω + = + + = + + = + = The let s proe the correctess of GRN-OEL- ALGORITHM Theore : Let ω be a geeralzed rg core etwor f N ad 0(od 4) the the GRN-OEL- ALGORITHM deteres a odd-elegat labelg of ω Proof: For eery geeralzed rg core etwor ω f N ad 0(od 4) t has = p = + odes ad has q = edges Let + = 0 ad classfy the odes of ω to three sets V V V the for eery ode of three sets label accordg to the labelg = fucto f proposed the GRN-OEL- ALGORITHM : = + f( ( j ) ) = + ( j ) [ ] j [ ]; 4 = + + f( ( j ) ) = + j [ ] j [ + ]; 4 = + f( ) j [ ] j [ ] + ( j) = + = + Frst accordg to the labelg fucto f we are able to calculate all ode labelg of ω : + = + 4 = + = = = f( V ) { ( j ) [ ] j [ ]} = = + f( V ) { j [ ] j [ ]} = = = { } = = = { = = = } + = + f( V ) = { + j [ ] + = } = + = j [ ] } = { = = = = + ; = = 864

4 Let the ode label sets of geeralzed rg core etwor ω s f( V ) the f( V) = f( V ) Easy to ow fro expressos f( Vj ) j [ ] that other ay two sets are dsjot sets except f( ) = f( + ) ( [ ] ) uber set f( V ) ad f( V ) So label fucto f s jecte fro ( + )( ) [0 q ] = odes set V to uber set [0 ] = of geeralzed rg core etwor ω Secod for e = u E( G) u uder label fucto f let ge ( ) f( u) f( ) od q [ ] we hae: g( ) = f( ) + f( ) (od q) = + = + j= = + the for Therefore we hae ge ( ) = ge ( ) = {5 q } So labelg fucto f deteres a odd-elegat labelg of geeralzed rg core etwor ω Therefore GRN-OEL- ALGORITHM ca detere a odd-elegat labelg of ω Obously we ca get the followg cocluso fro theore : Theore Geeralzed rg core etwor ω s a odd-elegat etwor topology for N 0(od 4) = j The pleetato of odd-elegat labelg algorth of geeralzed rg core etwor ω Usg Matlab laguage we cople the GRN-OEL- ALGORITHM progra We fsh the label experet for sxty-eght geeralzed rg core etwor ω of p ode (p= ) Lted space oly the results of the label of geeral rg core etwor ω 884 ω 4 ω 48 ω 4888 ω 886 ω 86 ω ω s ge here (show fgure to fgure 9) The experet statstcs CPU te of labelg ad appg for aboe eghtee geeralzed rg core etwor ω ad aalyze deelopet tred of CPU te of labelg ad appg for geeralzed rg core etwor ω Statstc ad aalyss results of experetal are show table ad fgure 0 fgure Fg Odd-elegat labelg of ω 884 Fg Odd-elegat labelg of ω 4 Fg 4 Odd-elegat labelg of ω 48 Fg 5 Odd-elegat labelg of ω

5 Fg 6 Odd-elegat labelg of ω 886 Fg 7 Odd-elegat labelg of ω 86 Fg 8 Odd-elegat labelg of ω Fg 9 Odd-elegat labelg of ω TABLE CPU TIME STATISTICS OF ODD-ELEGANT LABELING AND MAPPING LABELING OF GENERALIZED RING CORE NETWORK G = ω ertex uber of G geeralzed rgcore etwor G CPU te of labelg of G CPU te of appg G 8 ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω ω

6 x Fg 0 The CPU te tred chart of odd-elegat labelg of geeralzed rg core etwor ω Fg The CPU te tred chart of appg odd- elegat labelg graph of geeralzed rg core etwor ω Accordg to the chage tred of fgure 0 we fd that fucto relato betwee CPU te of labelg for geeralzed rg core etwor ω ad ertex uber p of geeralzed rg core etwor ω s a approxate lear relatoshp We also fd that fucto relato betwee CPU te of appg odd-elegat labelg graph of geeralzed rg core etwor ω ad ertex uber p of geeralzed rg core etwor ω s a approxate lear relatoshp by fgure Therefore we ay safely draw the cocluso that the GRN-OEL- ALGORITHM s effecteess Cocluso The coputer pleetato of odd-elegat labelg algorth of specal etwor topology structure has practcal gudg sgfcace to coputer coucato etwor syste desg of fuctoal relablty low coucato cost I ths paper the author defes geeralzed rg core etwor ω desgs the GRN-OEL- ALGORITHM whe 0(od 4) proes odd-elegat of ths etwor topology wors out the correspodg software Acowledget Ths research was supported copletely by the Natoal Natural Scece Foudato of Cha uder Grat No No ad No 6607; Research Projects of Gasu Proce Educato Scece "Twelfth Fe" project No GS[05]GHB074 ad No GS[0]GHB090 Refereces [] G J Chag D F Hsu ad D G Rogers Addte aratos o a graceful thee:soe results o haroous ad other related graphs Cogr Nuer 98 pp8-97 [] X-Q Zhou BYao ad X-E Che Eery lobster s odd- elegat Iforato Processg Letters (0) 0- [] MZYoussef OE-cordal labelg Ars Cob 04 pp7-79 [4] X-Q Zhou BYao X-E Che ad H-X Tao A proof to the odd-gracefuless of all lobsters Ars Cob 0 (0)

7 [5] WFeg ad CXu A surey of the gracefuless of dgraphs It J Pure Appl Math 69 (0) 45-5 [6] MHussa ET Basoro ad KAl O super atagc total labelg of Harary graph Ars Cob 04 (0) 5- [7] ZGao Odd graceful labelgs of soe uo graphs Nat Sc Helogjag U 4 (007) 5-9 [8] C Barretos Odd-graceful labelgs of trees of daeter 5 Graphs Cob 009 pp