found that now considerable work has been done in this started with some example, which motivates the later results.

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1 8 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December A Study o Adjacecy Matrix for Zero-Divisor Grahs over Fiite Rig of Gaussia Iteger Prajali, Amit Sharma ad R.K.Vats, Deartmet of Mathematics, Natioal Istitute of Techology, Hamirur, 775, INDIA rajalishrma@rediffmail.com, asharmait@gmail.com, ramesh_vats@rediffmail.com ABSTRACT : The aer studies the characterizatio of ad Livigsto associate to a commutative rig with uity a adjacecy matrix corresodig to zero-divisor grahs of grah ΓR, whose vertex was Z(R) = Z(R) {}. The Zero fiite commutative rig of Gaussia iteger uder modulo divisor grahs also have discussed ad studied for semi. For each ositive iteger we calculate umber of zero- grous by De Meyer [4]. Redmod has geeralized the divisors & examie ature of the matrix, ad the we otatio of zero divisor grahs. O studyig this article it is geeralized the order of matrix i each case. Firstly, we have foud that ow cosiderable work has bee doe i this started with some examle, which motivates the later results. directio. Some time the zero divisor grah for R is allowed The study is useful i comuter sciece alicatio such as: to have as a vertex, i such case has a edge to every codig theory, etwork commuicatio, museum guard other vertex i grahs. For simlificatio, we have used the roblems, etc. defiitio excludig as a vertex. I the first istace of Keywords: Gaussia Iteger, Zero-divisor, Adjacecy this aer we cosider some fiite rigs of Gaussia iteger Matrix, Commutative rig. ad discuses ature of adjacecy matrix i each case AMS Classificatio: 5Cxx; 4C; 5Axx,3Axx. deedig o we also ivestigate that what ca be the order of matrix i each case, we have started with some. INTRODUCTION examle which illustrates the geeral results. The defiitio Let R be fiite commutative rig of Gaussia of adjacecy matrix for zero divisor grahs is as follows: iteger, Gaussia iteger cotais set of all comlex umbers a+ib, where a ad b are iteger. It is deoted by a i j Z[i]. It forms Euclidia domai uder usual comlex oeratios, with orm N(a+ib) = a +b. It is clear that a+ib is uit i Z[i] iff N(a+ib) =, which imlies that, -, i, -i are oly uits. Let <> be the ricial ideal geerated by i Z[i], where is a atural umber ad let Z={,,,3,4.-} be rig of iteger modulo. The factor rig Z[i]/<> is isomorhic to Z[i] =, If v i & v j rereset zero divisor,, Otherwise where vi ad vj are vertices of grah G. Oe more advatage of the grah that it also detects the ilotet elemet of idex, whe self loo foud. We will take basic defiitio from grah theory [5]-[6] for commutative rig with uity [7]. To avoid trivialities whe G has o edges, we will assume whe ecessary R is ot Itegral domai, (e., we left case 3(mod4) [4]). We {a+ib : a,b Z} which imlies that Z[i] is Pricial ideal rig. Therefore Z[i] is rig of Gaussia iteger uder modulo. study for the followig rigs Z [i], Z [i], Z q [i]. Some Cosider Z[i], ad let Z(R) be set of zero divisors of Z[i] ad examles are givig below for each case of Z [i] : G be zero divisor grah of Z[i]. Cosider a, b Z[i], the a ad b are said to be adjacet if a.b = b.a =. The rig of Gaussia I. (mod4) II. (mod4) iteger Z[i] forms field if 3(mod4) [], therefore the grah G has o edge if 3(mod4). The cocet of zero divisors grah was give by I. Beck [] but his motive was i colorig of grahs. I [] Aderso Ad for rig Z [i], > the cases are such as: I. (mod 4) II. 3(mod 4)

2 9 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December III. (mod 4) ad Last case for Z[i], whe =.q, (mod 4). The rig Z[i], Case.: Whe (mod4), e., R = Z[i] Firstly we move to discuss about R = Z[i] The set of zero divisor Z(R) = {+i}, the ossible edge is 8 8 self loo. Observatio from matrix: Grah for Z(R) is give as: The matrix corresodig to zero divisor grah of Z5[i] is o sigular. All vertices have self loo so trace of the matrix is 8. Fig. : Zero divisor grah of the Z[i] i The rak of the matrix is 8, therefore zero is ot the eige value of the above matrix. ad the adjacecy matrix is give as: Case..: For R = Z3[i]. Observatio from matrix: The set of zero divisor = {+5i, +8i, +3i, +i, 3+i, Rig has oly oe zero divisor 3+i, 4+6i, 4+7i, 5+i, 6+4i, 6+9i, 7+4i, 7+9i, 8+i, 8+i, Matrix is of order ad havig trace ad 9+6i, 9+7i, +i, +i, +3i, +i, +5i, +8i} determiat equal. The grah is show below: Case.: Whe (mod4), e., R = Z5[i] ad R=Z3[i]. Case..: For R = Z5[i]. The set of zero divisor Z(R) = {+i, 3+i, 4+i, +4i, +3i, +i, 4+3i, 3+4i} The grah is show below: 4+3i +i 3+i 4+i +3i +i +4i 3+4i +5i 5+i 8+i 3+i +i +3i 6+4i 4+7i 9+7i 6+9i +8i +i 5+i +5i +8i +3i +i 3+i 4+6i 7+4i 7+9i 9+6i 8+i +i Fig. 3: Zero divisor grah of the Z3[i] The adjacecy matrix will be of order 4 4 ad ca be Fig. : Zero divisor grah of the Z5[i] The matrix corresodig to zero divisor grah of Z5[i] costructed by usig defiitio. Observatio from matrix: The matrix corresodig to zero divisor grah of Z5[i] is o sigular. Trace of the matrix will be 4.

3 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December i Zero will ot be the eigevalue of matrix Case 3.: Whe 3(mod4), e., R = Z9[i], Z7[i] as rak of adjacecy matrix is 4. Let R = Z9[i], the set of the zero divisor is Z(R) = {3, 6, 3i, Theorem.: Let Z[i] be rig of Gaussia iteger uder 6i, 3+3i, 6+3i, 6+6i, 3+6i} modulo. Cosider Z[i], (mod4), be rime. Let M The corresodig zero divisor grah is rereseted as: be adjacecy matrix corresodig to zero divisor grah of Z[i]. The order of adjacecy matrix i such case is always (-) (-). 6+3i 3+6i 6+6i Proof: The rig of Gaussia iteger Z[i] is fiite 3 commutative rig uder modulo. It is kow by theorem [] that i a fiite commutative rig each o-zero elemet 3i is either a uit or a zero divisor. To obtai umber of zero divisor we subtract uits from o zero elemet. It is foud 6 3 that uits i Z[i] are φ() φ(), therefore o. of zero divisor = -(-) (-)- = (-) Fig. 4: Zero divisor grah of the Z9[i] Thus it have roved that order of matrix is (-) (-). Theorem.: Let Z[i] be rig of Gaussia iteger uder Matrix for the above figure is modulo. Cosider Z[i], (mod4), be rime. Let M be adjacecy matrix corresodig to zero divisor grah of Z[i]. The adjacecy matrix will always be o sigular. examle it has observed that if a+ib rereset zero divisor the b+ia also rereset zero divisor. From the grah it is are coected, thus i adjacecy matrix o two rows are idetical. Therefore we have show that matrix will be o 8 8 Observatio from matrix: The matrix for the zero divisor of Z9[i] is sigular. Trace of the adjacecy matrix is 8, as all the sigular. Theorem.3: Let Z[i] be rig of Gaussia iteger uder modulo. Cosider Z[i], (mod4), be rime. Let M be adjacecy matrix corresodig to zero divisor grah of Z[i]. The trace of adjacecy matrix is always equal to umber of zero divisor. Proof: Let us cosider Z[i], (mod 4) as above discussed foud there is o such vertex at which a+ib ad b+ia both vertices have self loo. i Rak of the matrix is. iv. Eige values of above matrix are ad 8. v. Adjacecy matrix corresodig to Z(R) is diagoalizable. Theorem 3.: Let Z[i] be rig of Gaussia iteger uder Proof: Let us cosider Z[i], (mod 4) as above modulo. Cosider Z [i], 3(mod4), be rime. Let discussed examle it has observed that if a+ib rereset zero M be adjacecy matrix corresodig to zero divisor grah divisor the b+ia also rereset zero divisor. From the grah it is foud that all the vertices has self loo, therefore trace is of Z [i], the order of adjacecy matrix i such case is equal to umber of zero divisor. always The rig Z [i] Proof: Similar as theorem. -.

4 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December Theorem 3.: Let Z[i] be rig of Gaussia iteger uder modulo. Cosider Z [i], 3(mod4), be rime. Let Observatio from matrix: M be adjacecy matrix corresodig to zero divisor grah of Z [i]. The trace of adjacecy matrix is always atural umber >. or - - as well as i, i, i grah of Z4[i] is sigular. The trace of above matrix is 3. i Rak of adjacecy matrix is 3, which is less tha seve so zero must be eigevalue. 3 Proof: Let us cosider Z [i], 3(mod 4), here,, rereset zero divisor with The Adjacecy matrix with resect to zero divisor Case 3.. for R = Z8[i] The set of zero divisor Z(R) = {, 4, i, 4i, 6, 6i, +i, 4+4i, itself, e., grah must have self loo at least two airs which 7+5i 5+7i are cojugate. Therefore, at least the diagoal etry, e., aii ad ajj of the adjacecy matrix cotai. Thus the trace of +4i 4+i 5+3i 3+5i +i 6+6i +i matrix is atural umber, >. Case 3.: Whe (mod4), e., R = Z4[i], Z8[i] Case 3..for R = Z4[i], the set of zero divisors for Z4[i] = {, i, +i, 3+i, +i, +3i, 3+3i} The ossible edges for the grah are {,}, {i,i}, {+i,+i}, {+i,+i}, {+i,3+i}, {,i}, {,+i}, 5+5i 3+7i i 3+i +3i 7+i +7i +5i 5+i 6+4i 4+6i 4+4i +6i 6+i 3+3i 4i 6i 4 6 i {i, +i}, {3+3i,+i}, {+i,+3i} +5i, +7i, 5+i, 3+i, +i, 3+3i, +6i, 6+i, 7+7i, 4+6i, Grah is give as 6+4i, 7+i, 5+7i, 7+5i, 5+5i, 4+i, +4i, 3+5i, 5+3i, 4+4i, +i +3i, 3+7i, 7+3i} ad grah of the zero divisor is give as: 3+i +i +3i i Fig.6: Zero divisor grah of the Z8[i] ad the matrix for the zero divisor grah will be of order 3 3 ad ca be costructed i similar maer. 3+3i Observatio from matrix: grah of Z8[i] is sigular. Fig. 5: Zero divisor grah of the Z4[i] The adjacecy matrix for the grah: The Adjacecy matrix with resect to zero divisor The trace of above matrix is 3. i Rak of adjacecy matrix is. modulo. Cosider Z [i], (mod4), be rime. Let M be adjacecy matrix corresodig to zero divisor grah Theorem 3.3: Let Z[i] be rig of Gaussia iteger uder 7 7 of Z [i]. The order of adjacecy matrix i such case is always.

5 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December Proof: Similar as theorem as the uits of Z [i], are 5+5i by usig the referece []. +i 3+i +3i 4i i Theorem 3.4: Let Z[i] be rig of Gaussia iteger uder modulo. Cosider Z [i], (mod4), be rime. Let 3+3i 3i M be adjacecy matrix corresodig to zero divisor grah 4+4i 3 +5i 3+5i 5+3i 4+i +4i +i 5+i 4 of Z [i]. The adjacecy matrix i this case is always sigular. Fig. 7: Zero divisor grah of the Z6[i] Proof: Let us cosider Z [i], (mod4) as above The adjacecy matrix is of order 9 9 ad ca be discussed examle it has observed that if a+ib rereset zero costructed similarly. divisor the b+ia also rereset zero divisor. From the grah, it Observatio from matrix: is foud that at least two vertices ( -)(+i) ad (+i) rereset zero divisor with ( i). I adjacecy matrix at least two rows will be idetical, thus determiate The determiat of the matrix is zero. Trace of the adjacecy matrix is. i Rak of the matrix is 4. iv. Matrix is sigular so, zero must be eige value. of matrix is zero. Theorem 4.: Let Z[i] be rig of Gaussia iteger uder Theorem 3.5: Let Z[i] be rig of Gaussia iteger uder modulo. Cosider Z [i], (mod4) be rime. Let M be adjacecy matrix corresodig to zero divisor grah of Z [i]. The trace of matrix i such case is always three. Proof: Let us cosider modulo. Cosider Z [i], (mod4), be rime. Let M be adjacecy matrix corresodig to zero divisor grah of Z [i]. The adjacecy matrix i this case is always sigular. Z [i], (mod4) as above discussed examle it has observed that if a+ib rereset zero Z [i] Proof: Let R = be rig over Gaussia iteger (mod4). If a+bi rereset zero divisor the b+ai also divisor the b+ia also rereset zero divisor. From the grah i it is foud that vertices, ad ( i) which gives zero divisor, ad the vertices (a +bi) ( i) = roduces self loo always. Thus trace of adjacecy matrix is ad (b+ia) three []. due to above roduct will be idetical so determiat of ( i) =, e., some of rows of matrix matrix is zero. 4. The rig Z[i] Whe =.q, (mod4), where ad q are distict rime Theorem 4.: Let Z[i] be rig of Gaussia iteger uder umbers, e., R = Z6[i], Z[i], Case 4. for R = Z6[i], modulo. Cosider Z [i], (mod4), be rime. Let Cosider Z6[i], the set of zero divisors Z(R)= {, 3, 4, i, 3i, M be adjacecy matrix corresodig to zero divisor grah 4i, 3+3i, +i, +i, 4+4i, +4i, 3+i, 5+i, +3i, 5+3i, 5+5i, of Z [i]. The trace of adjacecy matrix is always 3+5i, 4+i, +5i}. The zero divisor grah is show as: atural umber, k>.

6 3 Iteratioal Joural of Comuter Sciece & Emergig Techologies (E-ISSN: 44-64) Volume, Issue 4, December [3] D.F. Aderso, P.S. Livigsto, The Zero-divisor Proof: Let R = Z [i], let (mod4), e., form of this case vertex i ( i) reresets zero divisor ad i fact, whe grah is formed vertex (999). [4] F.R.DeMeyer, T.Mckezie, K.Scheider, The Zerodivisor Grah of a Commutative Semi-grous, Semi ( i) always have self loo, whe adjacecy matrix is costructed the Grah of Commutative Rig, Joural of Algebra 7, Grou Forum 65, 6-4(). [5] R.Diestel, Grah Theory, Sriger-Verlag, Newyork, 977. ( i) vertex have etry (diagoal form), so trace is at least k, k>. [6] F. Harary, Grah Theory, Addiso-Wesley, Readig, MA, 97. [7] I.Kalasky, Commutative Rigs, Uiv. of Chicago 5. Coclusio: I this aer, we study adjacecy matrices for zero divisor Press, Chicago, 974. [8] V.K Bhat, Ravi Raia, A Note o Zero-divisor Grah grah over fiite rigs of Gaussia iteger. Grahs are the over Rigs, It. J. Cotem. Math. Sc (4), 667- most ubiquitous models of both atural ad huma made 67(7). structures. I comuter sciece, zero divisor grahs are used to rereset etworks of commuicatio, etwork flow, [9] B.Bollabs, Grah Theory-A Itroductory Course, Sriger-Verlag, Newyork, 979. clique roblems. Art gallery ad museum guard roblem are [] M.F Atiyah, I.G, Macdoald, Itroductio to a well-studied visibility roblem i comutatioal geometry. Commutative Algebra, Addiso-Wesley, Readig, MA, 989. REFERENCES [] J. A. Gallia, Abstract Algebra, Narosa Publishig House, ISBN: (998). [] I. Beck, Colorig of Commutatig Rig, J. Algebra 6, 8-6(988). [] Nafiz Abu Jaradeh Emad Abu Osba ad Salah AiAddasi, Zero Divisor grah for the rig of Gaussia itegers modulo, Taylor & Fracis, Commuicatios i algebra, 36: (8).