International Journal of Scientific & Engineering Research, Volume 6, Issue 9, September ISSN

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1 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 9 ISSN Doinating Sets Of Divisor Cayley Grahs G. Mirona, Prof. B. Maheswari Abstract Let n be an integer and S be the set of divisors of n.then the set S*= {s, n- s / s S, n s} is a syetric subset of the grou (Zn, ), the additive abelian grou of integers odulo n. The Cayley grah of (Zn, ), associated with the above syetric subset S* is called the Divisor Cayley grah and it is denoted by G(Zn, D).That is, G(Zn, D) is the grah whose vertex set is V={0,,,,n-} and the edge set is E = {(x, y) / x-y or y-x is in S*}. Let G (V,E) be a grah. A subset D of V is said to be a doinating set of G if every vertex in V \ D is adjacent to a vertex in D. A doinating set with iniu cardinality is called a iniu doinating set and its cardinality is called the doination nuber of G and is denoted by ϒ (G). Index Ters Divisor Cayley Grah, doinating set, doination nuber.. INTRODUCTION Berge[4] and Ore[] are the first to introduce the concet of doination and they have contributed significantly to the theory of doination in grahs. E.J. Cockayne and S.T. Hedetniei [6] ublished the first aer entitled otial doination in grahs. They were the first to use the notation γ (G) for the doination nuber of a grah, which subsequently becae the acceted notation. Allan and Laskar [], Cockayne and Hedetniei [6], Aruuga [], Saath Kuar [] and others have contributed significantly to the theory of doinating sets and doination nubers. An introduction and an extensive overview on doination in grahs and related toics are given by Haynes et al. [8]. In the sequel edited by Haynes, Hedetniei and Slater [9], several authors resented a survey of articles in the wide field of doination in grahs. Cockayne, C.J et.al [5] introduced the concet of total doination in grahs and studied extensively. The alications of both doination and total doination are widely used in Networks. In this chater we discuss doinating sets of Divisor Cayley Grahs. Here we have obtained these sets for various values of n. In certain cases we could get inial doinating sets. In other cases, we are unable to give the roofs to find the inial sets by the technique we adoted to find these sets. This is because, the divisors deend on the nuber n and unless we give the value for n, we are not known the divisors and hence the eleents in S *. Hence we have devised an algorith which finds all inial doinating and total doinating sets for all n. Our algorith finds all closed and oen neighborhood sets resectively and then finds a iniu nuber of these sets which cover all the vertices of G(Zn, Each theore is strengthened with exales and the Algorith is also illustrated for different values of n. DOMINATING SETS OF DIVISOR CAYLEY GRAPHS Let us define a doinating set as follows Definition : Let G (V,E) be a grah. A subset D of V is said to be a doinating set of G if every vertex in V \ D is adjacent to a vertex in D. A doinating set with iniu cardinality is called a iniu doinating set and its cardinality is called the doination nuber of G and is denoted by γ(g). We now find doinating sets of Divisor Cayley grah. Theore.: If n is a rie, then D = {rd0/0 r k-, where k is largest ositive integer such that rdo < n and do = }, is a doinating set of G (Zn, Proof: Let n be a rie. Then G (Zn, D), is an outer Hailton cycle. Let D = {rd0/0 r k-, where k is the largest ositive integer such that rdo< n, do = }. If we construct D, then the vertices of D are in the for {0,,6,9,.}. Since G (Zn, D) is an outer Hailton cycle, each vertex in D is adjacent with its receding and succeeding vertices. This ilies that every vertex in V-D is adjacent with at least one vertex in D. Therefore, D becoes a doinating set of G (Zn, Also by the construction of D, it is clear that D is inial since no roer subset of D is a doinating set and hence D = k is the doination nuber of G (Zn, Exale. : Consider G (Z7, D) for n = 7 which is a rie. Then S ={,7} and S * = {,6}. The grah G (Z7, D) is regular. The grah of G (Z7, D) is as follows. Det. of Alied Matheatics, SPMVV, Tiruati, A.P. India Eail:aherahul.55@gail.co 05

2 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 0 ISSN ,, -, - 4, +,, , + 6, 5, Also, G (Z7, D) is an outer Hailton cycle. If we construct D, we get D = {0,,6}. We observe that the set D = {0,,6} doinates all the vertices in V D. Further, this D is a inial doinating set because if we delete one vertex fro this set then the reaining vertices cannot doinate the vertices of G(Z7, D). Therefore, the doination nuber of G (Z7, D) is. A ossible nuber of MDSs of G (Z7, U7) is 7 and these sets are given by {0,,6}, {,4,0}, {,5,}, {,6,}, {4,0,}, {5,,4}, {6,,5}. Reark : The doination nuber of G(Zn, D) is if n =,,4 and 6. The syetric subset S * contains all the vertices of G(Zn, D) for all value of n and hence the difference of any two vertices is in S *. Thus every vertex is adjacent to all the vertices of G(Zn, D) so that it is colete. So, every single tie vertex is doinating set and hence the doination nuber is. Theore. : Let n =, be a rie. Then D = {r d0 / 0 r k- where k is the largest ositive integer such that rd0 < n and d0= } is a doinating set of G(Zn, Proof : Consider the grah G(Zn, Let n =, be a rie. In this case, S = {,, ) and S* = {,, n-, n-}. i.e., S* = {,, -, -}. The grah G(Zn, D) is S* - regular. That is the grah is 4 - regular. Let D = {rd0 / 0 r k- where k is the largest integer such that rd0 < n and d0 = }. We now show that D is a doinating set of G(Zn, Clearly, the vertex 0 is adjacent to the vertices,, -, - as their difference is resectively,, -, - which belongs to S *. The vertex is adjacent to the vertices 4, +,, - +. The vertex 6 is adjacent to the vertices 7, + 6, 5, This is exressed in the following. Here we observe that the vertices {0,, 6, 9,,. } are doinating the vertices 0,,,, 4, 5, 6, 7.. If we give the value for then the rest of the vertices which are doinated by {0,, 6, 9, } will also be found. Likewise the vertices in D doinate the vertices of the grah G(Zn, D) and hence D is a doinating set of G(Zn, Exale.4: Let n = 9. Then S = {,,9} and S * = {,,6,9}. The grah of G (Z9, D) is as follows. Then D becoes {0,, 6} and the vertices adjacent to D are 0,, 8, 6 4, 6,, 0 6 7, 0, 5, i.e., the set {0,, 6} doinates the vertices {0,,,, 4, 5, 6, 7, 8, 9}. Hence D = {0,, 6} becoes a doinating set of G(Z9, This set is also inial. If we delete any one vertex fro D, then the resulting set will not be a doinating set of G(Z9, D) since the grah is 4 regular, a set of vertices cannot doinate all the vertices of the grah. Exale.5: Suose n = 5. Then S = {,5,5} and S * = {,5,0,4}. The grah of G (Z5, D) is as follows. 05

3 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 ISSN Now D = {0,, 6, 9,, 5, 8, }. The vertices adjacent with the vertices of D are 0, 5, 4, 0 4, 8,, 6 7,, 5, 9 0, 4, 8, 4, 7,, 7 5 6, 0, 4, 0 8 9,, 7,,, 0, 6 Here we observe that the set D = {0,, 6, 9,, 5, 8, } doinates all the vertices of the grah G(Z5, Since the grah is 4-regular we should have D 7. But here we have obtained D = 8. That is D is not inial. To get inial doinating sets, we give the following algorith which finds the inial doinating sets of G(Zn, If we run the algorith, then the doinating set obtained for this grah is D = {0,,, 4, 7,, } whose cardinality is 7. This algorith is run by finding a iniu set of closed neighbourhood sets which cover all the vertices of G(Zn, The following algorith finds inial doinating sets of G(Z n, D) for all values of n excet when n is a rie. Because when n is a rie, the grah becoes an outer Hailton cycle and hence the doinating sets are found easily. Algorith DS - DCG: INPUT : Enter a nuber OUTPUT : Minial doinating sets STEP : Enter a nuber n STEP : IF n IS NOT EQUAL TO NULL GOTO STEP ELSE GOTO STEP STEP : METHOD setofdivisiors($nuber){ INITIALIZE ARRAY $divisorset FOR EACH $i,$i=,...$i<=$nuber DO THE FOLLOWING INITIALIZE $odval ASSIGN $nuber % $i TO $odval IF $odval EQUAL TO 0 ASSIGN $i TO $divisorset[] RETURN $divisorset STEP 4 : Find set of divisors CALL METHOD setofdivisiors(ara) INITIALIZEVARIABLE$resultASSIGN COUNTOF(setofdivisiors (ara)) INITIALIZE VARIABLE $setofdivisorsarray ASSIGN setofdivisiors(ara) PRINT "Set of divisors ={"; FOR EACH $i,$i=0,...$i<ara INITIALIZEARRAY$setOfDivisorsArrayTe[]ASSIGN $setof DivisorsArray[$i]; PRINT $setofdivisorsarray[$i]; IF $i NOT EQUAL TO $result- PRINT ","; PRINT "}" STEP 5 : Find Syetric sub set INITIALIZEVARIABLE$setOfDivisorsCount ASSIGNCOUNTOF($setOfDivisorsArrayTe); INITIALIZE ARRAY $firsttearray INITIALIZE ARRAY $secondtearray FOR EACH $j,$j=0,...$j<$setofdivisorscount IF $setofdivisorsarrayte[$j] NOT EQUAL TO $nuber IF $setofdivisorsarrayte[$j] EQUAL TO $nuber BREAK ELSE ASSIGN VALUE $setofdivisorsarrayte[$j] TO $firsttearray[] ASSIGNVALUE($nuber$setOfDivisorsArrayTe[$j]) TO $second TeArray[] MERGE $firsttearray AND $secondtearray array_erge($firsttearray,$secondtearray) 05

4 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 ISSN INITIALIZE VARIABLE $ergefstes REMOVEDUPLICATEVALUESUSINGarray_unique (array_erge ($firsttearray, $secondtearray)) ASSIGN TO $ergefstes SORT ARRAY asort($ergefstes); INITIALIZE VARIABLE $keynu ASSIGN 0 TO $keynu WHILE (list($key, $val) = EACH($ergeFSTes)) { INITIALIZE ARRAY $syetricsubset ASSIGN VALUE $val TO $syetricsubset[] INCREMENT $keynu++; END WHILE INITIALIZE VARIABLE $syetricsubsetcount ASSIGN COUNT OF count($syetricsubset) TO $syetricsubsetcount PRINT "S* ={" FOR EACH $,$=0,...$<$syetricSubSetCount PRINT $syetricsubset[$]; IF $ NOT EQUAL TO $syetricsubsetcount- PRINT ","; STEP 7 : FOR PRINT "Consider N[0] ={"; PRINT "}" PRINT "}" STEP 6 : Find Neighbourhood sets of n STEP 8 : Find Uncovered vertices in N[0] FOR EACH $nsa,$nsa=0,...$nsa<$nuber FOR EACH $,$=0,...$<$nuber FOR EACH $nsb,$nsb=0,...$nsb<$syetricsubset INITIALIZE ARRAY $notin INITIALIZE MULTI DIMENSIONAL ARRAY IF!in_array($, $nof0array) $neighbourhoodsetarray ASSIGN$ TO $notin[] ASSIGN $syetricsubset[$nsb] TO $neighbourhoodsetarray[$nsa][$nsb] ASSIGN $syetricsubset[$nsb]+ TO $syetricsubset[$nsb] IF $syetricsubset[$nsb] EQUAL TO $nuber ASSIGN 0 TO $syetricsubset[$nsb] / /FIRST ELEMENTS UNION FOR EACH $ns,$ns=0,...$ns<$nuber PUSH TO FIRST PLACE array_unshift($neighbourhoodsetarray [$ns], $ns) PRINT "Neighbourhood sets of n are" FOR EACH $ns,$ns=0,...$ns<$nuber PRINT "N[$ns]={" FOR EACH $ nbs,$nbs=0,...$nbs<count ($neighbourhoodsetarray[$ns] PRINT $neighbourhoodsetarray[$ns][$nbs]; IF $nbs EQUAL TO count($neighbourhoodsetarray[$ns])- PRINT "" ELSE PRINT "," PRINT "}"; EACH $,$=0,...$<count($neighbourhoodSetArray[0]) INITIALIZE ARRAY $nof0array ASSIGN $neighbourhoodsetarray[0][$] TO $nof0array[] PRINT $neighbourhoodsetarray[0][$]; IF $ NOT EQUAL TO count($neighbourhoodsetarray[0])- PRINT ","; PRINT "Uncovered vertices in N[0] are {"; FOR EACH $,$=0,...$<count($notin) PRINT $notin[$]; IF $ NOT EQUAL TO count($notin)-) PRINT ",";. STOP Theore.6 : Let n =, where = and >. Then D= {rd0 / 0 r k- where k is the largest ositive integer such that rd0 < n and d0 = 5} is a doinating set of G(Zn, Proof : Let n = where = and >. Consider the grah G(Zn, In this case, S = {,,. } and S* = {,,., -, -, -, -,. - - }. The grah is S* - regular. 05

5 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 ISSN Let D = {rd0 / 0 r k- where k is the largest integer such that rd0 < n and d0 = 5}. i.e., D = {0, 5, 0,5,...}. We now show that D is a doinating set of G(Zn, The vertices in D are adjacent with the following vertices. 0,, -, -, -,., , +5, + 5,. + 4, + 5, - + 5, - + 5,., , +0, , +0, + 0, - +0,., If we relace = then the adjacency of the vertices is as follows. 0,,4,8,6,., - 5 6,7,9,,,., ,,4,8,6,,., If we give the value for, then the reaining set of vertices which are doinated by {0,5,0,5.} will be found. Likewise, the vertices in D doinate all the vertices of the grah G(Zn, D) and hence D is a doinating set of G(Zn, Exale.7 : Let n = 6 i.e., = and = 4. Then S = {,,4,8,6} and S * = {,,4,8,,4,5}. The grah of G (Z6, D) is as follows. Then D becoes {0, 5, 0, 5} and the vertices adjacent with the vertices of D are 0,, 4, 8,, 4, 5 5 6, 7, 9,,,, 4 0,, 4,, 6, 8, 9 5 0,,, 7,,, 4 i.e., {0, 5, 0, 5} doinates all the vertices of G(Z6, Hence this set becoes a doinating set of G(Z6, Since G(Z6, D) is 7-regular, we should have D. Here we have obtained D = 4. i.e., D is not inial. If we run the Algorith, then the inial doinating set obtained for this grah is D = {0, 5, 9} whose cardinality is. α α Theore.8 : If n = where and are ries such that α, α then D = {rd0 / 0 r k- where k is the largest integer such that rd0 < n and d0 = 5} is a doinating set of G(Z n, D). Proof : Consider the grah G(Zn, D). α α Let n = where and are ries and α, α. In this case, S = {,,,.., α,,, α,.,.,.,. α,.,..., α., n}. S* = {,,, α,,.., α,.,.,.,.. α,.,.,.,, α.,n-, n -, n -..,n - α, n -, n -,.., n - α, n -., n -., n.,, n. α, n -., n -. P,, n - α. α }. The set of vertices which are adjacent with the vertices in D is exressed in the following. 0,,, α,,.., α,.,.,.,.. α,.,.,, α., n-, n -, n -,..,n - α, n -, n -,.., n - α, n -., n -., n.,, n. α, n -., n -. P,, n - α. 5 6, + 5, + 5,, α + 5, + 5, +5,, α +5,. + 5, + 5,. + 5,. α + 5,. +5,. + 5,, α. + 5, n+4, n - + 5, n - + 5,., n- α +5, n-+5, n- +5,, n- α +5, n-. +5, 05

6 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 4 ISSN The grah of G (Z0, D) is as follows. n-. +5,, n- +5, n , + 0, + 0,, α + 0, + 0, +0,, α +0,. + 0,. + 0,. + 0,. α + 0,. +0,. + 0,, α. + 0, n+9, n - + 0, n - + 0,., n- α +0, n-+0, n- +0,, n- α +0, n-. +0, n-. +0,, n- +0, n Here we observe that the vertices {0, 5, 0,.} are doinating the vertices, 6,,.. If we give the values for,, α and α, then the set of vertices which are doinated by {0, 5, 0, 5.} will also be found and this set is nothing but the vertex set of G(Zn, Likewise, the vertices in D doinate the vertices of the grah G(Zn, D) and hence D becoes a doinating set of G(Z n, Exale.9 : For n =, where =, =, α =, α =, S = {,,,4,6,} and S * = {,,,4,6,8,9,0,}. The grah of G (Z, D) is as follows. Now D = {0, 5} and the vertices adjacent to D are 0,,, 4, 6, 8, 9, 0, 5 6, 7, 8, 9,,,,, 4 Thus D = {0, 5} doinates all the vertices of G(Z, D) and hence this set becoes a inial doinating set of G(Z, Exale.0: For n = 0 where =, = 5, α =, α =, S = {,,4,5,0, 0} and S * = {,,4,5,0,5,6,8,9}. Then D = {0, 5, 0, 5} and the vertices adjacent to D are 0,,4,5,0,5,6,8,9 5 6,7,9,0,5,0,,,4 0,,4,5,0,5,6,8,9 5 6,7,9,0,5,0,,,4 i.e., {0,5,0,5} doinates all the vertices of G(Z0, Hence this set becoes a doinating set of G(Z0, Since G(Z0, D) is 9-regular, we should have D. Here we obtained that D =4. i.e., D is not inial. By the Algorith, the inial doinating set of G(Z0.D) is {0, 7, 9} whose cardinality is. Theore. : Let n = α, α.. α where,.. are ries and α, α. α.then D = {rd0 / 0 r k- where k is the largest ositive integer such that rd0 < n and d0 = 6} is a doinating set of G(Zn, Proof : Let n = α, α.. α. In this case, S = {,,.., α,,,.., α,....,,..., α,.,.,...,. α,.,.., α. α,..., α α.. α }. S * = {,,.., α,,,.., α,....,,..., α,.,.,...,. α,.,.., α. α,..., α α.. α, n, n -,., n - α,, - n- α α -. }. The grah is S* - regular. Let D = {rd0 / 0 r k- where k is the largest ositive integer such that rd0 < n and d0 = 6}. We now show that D is a doinating set of G(Zn, 05

7 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 5 ISSN The set of vertices adjacent to the vertices of D are given below. 0,,.., α,,,.., α,....,,..., α,.,.,...,. α,.,.., α. α,..., α α.. α, n, n -,., n - α -,, n- α α , + 6,.., α +6., +6, + 6 α +6,. -., α α , n + 5, n- + 6,.., n- α + 6., -, n- + 6, n- + 6, n- α , +,.., α +., +,.., α +,, - α α , n +,.., n- α ,8,9,,,0,7,4,4,0,,4,5,4,5,8,9,6,,40,5,6,9,0, 8 9,0,,4,5,,9,4,,,5,6,7 4 5,6,7,0,,8,,0,7,8,,, 0,,,6,7,,9,6,,4,7,8,9 6 7,8,9,0,,8,5,,9,0,,4,5 i.e., D = {0,6,,8,4,0,6} doinates all the vertices of G(Z 4, Hence this set becoes a doinating set of G(Z4, Since (G(Z 4, D) is -regular, we should have D 4. Here we have obtained D = 7. i.e., D is not inial. By the Algorith, the inial doinating set of G(Z4, D) is {0,0,6,6,6} whose cardinality is 5. The given algorith is develoed by using the PHP software (server scriting language) and the following illustrations are obtained by sily giving the value for n.. Algorith - Illustrations. n =, be a rie. Let n =5 S ={,5,5} S* ={,5,0,4} Neighbourhood sets of G(Z5, D) are N[0]={0,,5,0,4} N[]={,,6,,0} N[]={,,7,,} N[]={,4,8,,} N[4]={4,5,9,4,} Here we observe that the vertices {0,6,,8,.} are doinating the vertices 0,,5,7, It we give the value for,,., and α, α,, α then the rest of the vertices which are doinated by {0,6,,.} will also be found. Likewise, the vertices in D doinate the vertices of the grah G(Z n, D) and hence D is a doinating set of G(Z n, N[5]={5,6,0,0,4} Exale. : Let n = 4. Here =, =, = 7, α = N[6]={6,7,,,5} α = α =. N[7]={7,8,,,6} Then S = {,,,6,7,4,,4} and S* N[8]={8,9,,,7} ={,,,6,7,4,,8,5,6,9,40,4}. N[9]={9,0,4,4,8} The grah of G (Z4, D) is as follows. N[0]={0,,5,5,9} N[]={,,6,6,0} N[]={,,7,7,} N[]={,4,8,8,} N[4]={4,5,9,9,} N[5]={5,6,0,0,4} N[6]={6,7,,,5} N[7]={7,8,,,6} N[8]={8,9,,,7} N[9]={9,0,4,4,8} Fig. 7 : G (Z4, D) N[0]={0,,0,5,9} Then D becoes {0, 6,, 8, 4, 0,6} and the vertices N[]={,,,6,0} adjacent with the vertices of D are N[]={,,,7,} 0,,,6,7,4,,8,5,6,9,40,4 05

8 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 6 ISSN N[]={,4,,8,} N[4]={4,0,4,9,} α α. n = Minial doinating set is ={0,,,4,7,,}. where and are ries such that The nuber of inial doinating sets of G(Z 5, D) α, α. are Let n =0 {0,,,4,7,,},{,4,,5,8,,},{,5,,6,9,,4},{,6, S ={,,4,5,0,0} 4,7,0,4,5}, S* ={,,4,5,0,5,6,8,9} {4,7,5,8,,5,6},{5,8,6,9,,6,7},{6,9,7,0,,7,8},{7, Neighbourhood sets of G(Z0, D) are 0,8,,4,8, 9}, N[0]={0,,,4,5,0,5,6,8,9} {8,,9,,0,9,0},{9,,0,,,0,},{0,,,4,,, N[]={,,,5,6,,6,7,9,0} }, {,4, N[]={,,4,6,7,,7,8,0,},0,,,},{,5,,,4,,4},{,6,4,,5,4,5}, N[]={,4,5,7,8,,8,9,,} {4,7,0,,6,5, N[4]={4,5,6,8,9,4,9,0,,} 6},{5,8,,4,7,6,7},{6,9,,5,8,7,8},{7,0,,6,9,8, N[5]={5,6,7,9,0,5,0,,,4} 9}, {8,, 4,7, N[6]={6,7,8,0,,6,,,4,5} 0,9,0},{9,,5,8,,0,},{0,,6,9,,,}, N[7]={7,8,9,,,7,,,5,6} {,4,7,0,, N[8]={8,9,0,,,8,,4,6,7},},{,0,8,,4,,4},{,,9,,5,4,0}. N[9]={9,0,,,4,9,4,5,7,8} {4,,0,,6,0,}. N[0]={0,,,4,5,0,5,6,8,9} The doination nuber is 7. N[]={,,,5,6,,6,7,9,0}. n = where = and >. N[]={,,4,6,7,,7,8,0,} Let n =6 N[]={,4,5,7,8,,8,9,,} S ={,,4,8,6} N[4]={4,5,6,8,9,4,9,0,,} S* ={,,4,8,,4,5} N[5]={5,6,7,9,0,5,0,,,4} Neighbourhood sets of G(Z6, D) are N[6]={6,7,8,0,,6,,,4,5} N[0]={0,,,4,8,,4,5} N[7]={7,8,9,,,7,,,5,6} N[]={,,,5,9,,5,0} N[8]={8,9,0,,,8,,4,6,7} N[]={,,4,6,0,4,0,} N[9]={9,0,,,4,9,4,5,7,8} N[]={,4,5,7,,5,,} Minial doinating set is ={0,7,9} N[4]={4,5,6,8,,0,,} The nuber of inial doinating sets of G(Z0, D) are N[5]={5,6,7,9,,,,4} {0,7,9},{,8,0},{,9,},{,0,},{4,,},{5,,4},{6,,5}, N[6]={6,7,8,0,4,,4,5} {7,4, 6}, N[7]={7,8,9,,5,,5,6} {8,5,7},{9,6,8},{0,7,9},{,8,0},{,9,},{,0,},{4, N[8]={8,9,0,,0,4,6,7},}, {5,,4}, {6,,5},{7,4,6},{8,5,7},{9,6,8}. N[9]={9,0,,,,5,7,8} The doination nuber is. N[0]={0,,,4,,6,8,9} 4. Let n = α. α α where,.. are N[]={,,,5,,7,9,0} ries and α, α., α N[]={,,4,0,4,8,0,} Let n =4 N[]={,4,5,,5,9,,} S ={,,,6,7,4,,4} N[4]={4,5,0,,6,0,,} S* ={,,,6,7,4,,8,5,6,9,40,4} N[5]={5,0,,,7,,,4} Neighbourhood sets of G(Z4, D) are Minial doinating set is ={0,5,9} N[0]={0,,,,6,7,4,,8,5,6,9,40,4} The nuber of inial doinating sets of G(Z6, D) are N[]={,,,4,7,8,5,,9,6,7,40,4,0} {0,5,9},{,6,0},{,7,},{,8,},{4,9,},{5,0,4},{6,,5},{7, N[]={,,4,5,8,9,6,,0,7,8,4,0,},0}, N[]={,4,5,6,9,0,7,4,,8,9,0,,} {8,,},{9,4,},{0,5,},{,0,4},{,,5},{,,6},{4,,7},{ N[4]={4,5,6,7,0,,8,5,,9,40,,,} 5,4,8}. N[5]={5,6,7,8,,,9,6,,40,4,,,4} The doination nuber is. N[6]={6,7,8,9,,,0,7,4,4,0,,4,5} 05

9 International Journal of Scientific & Engineering Research, Volue 6, Issue 9, Seteber-05 7 ISSN N[7]={7,8,9,0,,4,,8,5,0,,4,5,6} N[8]={8,9,0,,4,5,,9,6,,,5,6,7} N[9]={9,0,,,5,6,,0,7,,,6,7,8} N[0]={0,,,,6,7,4,,8,,4,7,8,9} N[]={,,,4,7,8,5,,9,4,5,8,9,0} N[]={,,4,5,8,9,6,,40,5,6,9,0,} N[]={,4,5,6,9,0,7,4,4,6,7,0,,} N[4]={4,5,6,7,0,,8,5,0,7,8,,,} N[5]={5,6,7,8,,,9,6,,8,9,,,4} N[6]={6,7,8,9,,,0,7,,9,0,,4,5} N[7]={7,8,9,0,,4,,8,,0,,4,5,6} N[8]={8,9,0,,4,5,,9,4,,,5,6,7} N[9]={9,0,,,5,6,,40,5,,,6,7,8} N[0]={0,,,,6,7,4,4,6,,4,7,8,9} N[]={,,,4,7,8,5,0,7,4,5,8,9,0} N[]={,,4,5,8,9,6,,8,5,6,9,0,} N[]={,4,5,6,9,0,7,,9,6,7,0,,} N[4]={4,5,6,7,0,,8,,0,7,8,,,} N[5]={5,6,7,8,,,9,4,,8,9,,,4} N[6]={6,7,8,9,,,40,5,,9,0,,4,5} N[7]={7,8,9,0,,4,4,6,,0,,4,5,6} N[8]={8,9,0,,4,5,0,7,4,,,5,6,7} N[9]={9,0,,,5,6,,8,5,,,6,7,8} N[0]={0,,,,6,7,,9,6,,4,7,8,9} N[]={,,,4,7,8,,0,7,4,5,8,9,0} N[]={,,4,5,8,9,4,,8,5,6,9,0,} N[]={,4,5,6,9,40,5,,9,6,7,0,,} N[4]={4,5,6,7,40,4,6,,0,7,8,,,} N[5]={5,6,7,8,4,0,7,4,,8,9,,,4} N[6]={6,7,8,9,0,,8,5,,9,0,,4,5} N[7]={7,8,9,40,,,9,6,,0,,4,5,6} N[8]={8,9,40,4,,,0,7,4,,,5,6,7} N[9]={9,40,4,0,,4,,8,5,,,6,7,8} N[40]={40,4,0,,4,5,,9,6,,4,7,8,9} N[4]={4,0,,,5,6,,0,7,4,5,8,9,40} Minial doinating set is ={0,0,6,6,6} The nuber of inial doinating sets of G(Z4, D) are {0,0,6,6,6},{,,7,7,7},{,,8,8,8},{,,9,9,9},{ 4,4,0,0, 0}, {5,5,,,},{6,6,,,},{7,7,,,},{8,8,4,4, 4}, {9,9,5,5,5}, {0,0,6,6,6},{,,7,7,7},{,,8,8,8}, {,,9,9,9}, {4,4,40,0,0},{5,5,4,,},{6,6,0,,}, {7,7,,,}, {8,8,,4,4},{9,9,,5,5},{0,0,4,6,6}, {,,5,7,7}, {,,6,8,8}, {,,7,9,9},{4,4,8,40,0}, {5,5,9,4,}, {6,6,0,0,},{7,7,,,},{8,8,,,4}, {9,9,,,5},{0,40,4,4,6},{,4,5,5,7},{,0,6,6, 8}, {,,7,7,9},{4,,8,8,40},{5,,9,9,4},{6,4,0,0,0},{ 7,5,,,}, {8,6,,,},{9,7,,,},{40,8,4,4,4},{4,9,5,5,5}. The doination nuber is 5.. REFERENCES [] Allan, R.B. and Laskar, R.C.On doination and indeendent doination nubers of a grah, Discrete Math, (978), [] Aostol, M.To.Introduction to Analytic Nuber theory, Sringer International Student Edition, (989). [] Aruuga, S. Unifor Doination in grahs, National Seinar on grah theory and its alications, January (98). [4] Berge, C.The Theory of Grahs and its Alications, Methuen, London (96). [5] Cockayne, C.J., Dawes, R.M. and Hedetniei, S.T.Total doination in grahs, Networks, 0 (980), -9. [6] Cockayne, C.J., and Hedetniei, S.T.Towards a theory of doination in grahs, Networks, 7 (977), [7] Harary, F.Grah Theory, Addison Wesley, Reading, MA, (969). [8] Haynes, T.W., Hedetniei, S.T. and Slater, P.J.Fundaentals of doination in grahs, Marcel Dekker, Inc., New York (998). [9] Haynes, T.W., Hedetniei, S.T. and Slater, P.J.Doination in grahs: Advanced Toics, Marcel Dekker, Inc., New York (998). [0] Madhavi, L.Studies on doination araeters and enueration of cycles in soe Arithetic Grahs, Ph. D. Thesis, subitted to S.V. University, Tiruati, India, (00). [] Nathanson, B. Melvyn Connected coonents of arithetic grahs,monat. fur. Math, 9 (980), 9 0. [] Ore, O.Theory of Grahs, Aer. Math. Soc. Colloq. Publ. vol. 8.Aer. Math. Soc., Providence, RI, (96). [] Saath Kuar, E.On soe new doination araeters of a grah. A survey. Proceedings of a Syosiu on Grah Theory and Cobinatorics, Kochi, Kerala, India, 7 9 May (99), 7. [4] Sujatha, K.Sudies on Doination Paraeters and Cycle Structure of Cayley Grahs associated with soe Arithetical Functions, Ph. D. thesis, S.V.University, 008. [5] Vizing, V.G.Vertex colorings with given colors, Metody Diskret. Analiz. (in Russian) 9:-0,

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