On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo N

Open Journal of Discrete Mathematics, 2012, 2, 24-34 http://dx.doi.org/10.4236/ojdm.2012.21006 Published Online January 2012 (http://www.SciRP.org/journal/ojdm)

On the Line Graph of the Complement Graph for the Ring of Gaussian Integers Modulo n

Manal Ghanem1, Khalida Nazzal2 1Department of Mathematics, Irbid National University, Irbid, Jordan 2Department of Mathematics, Palestine Technical University (Kadoorie), Tulkarm, Palestine Email: [email protected], [email protected]

Received October 13, 2011; revised November 2, 2011; accepted November 22, 2011

ABSTRACT The line graph for the complement of the zero divisor graph for the ring of Gaussian integers modulo n is studied. The diameter, the radius and degree of each vertex are determined. Complete characterization of Hamiltonian, Eulerian, planer, regular, locally H and locally connected  is given. The chromatic number when n is a power Li n[] of a prime is computed. Further properties for and are also discussed. Li n[]  n[]i 

Keywords: Complement of a Graph; Chromatic Index; Diameter; Domination Number; Eulerian Graph; Gaussian Integers Modulo n; Hamiltonian Graph; Line Graph; Radius; Zero Divisor Graph

1. Introduction of Gaussian integers modulo n is investigated in [4]. In this paper it should be kept in mind that The line graph LG of a graph G is defined to be  Vi  [] ={1 i }, and hence, its line graph is K , the graph whose vertex set constitutes of the edges of G ,   2  0 Where two vertices are adjacent if the corresponding  q []i is an integral domain, so  q []iK = 0 . Further, edges have a common vertex in G . The importance of   []i is a complete graph whose complement is line graphs stems from the fact that the line graph  q2  transforms the adjacency relations on edges to adjacency totally disconnected and thus its line graph is K . While relations on vertices. For example, the chromatic index 0 of a graph leads to the chromatic number of its line graph.  p []iK = pp1, 1 , so its complement is disconnected The zero divisor graph of a commutative ring R , with two components each of which is isomorphic to denoted by R , is defined as the graph whose vertex K p1 . Finally, note that the graph  2q []i is bipartite, set is the set of all non-zero zero divisors of R and [1] and  qq []iK = 22. 12 qq121, 1 edge set ER= xyxyR : , 0 and xy = 0 . This In this paper, we investigate properties of the graph  type of graphs provides an example showing that algebraic Lin [] . We find the diameter, the radius of methods could be applied to problems about graphs. The Li [] . We determine which Li [] is Eu- set of Gaussian integers, denoted by []i , is defined as  n  n the set of complex numbers abi , where ab,   . If lerian, Hamiltonian, regular, locally H , locally connec- x is a prime Gaussian integer, then x is either ted or planer. Furthermore, the chromatic index and the 1) 1 i or 1 i , or edge domination number of  n[]i where n is a 2) q where q is a prime integer and q  3mod4, or power of a prime are computed. While the domination 3) abi , abi where ab22 = p, p is a prime number of  n[]i is given. On the other hand, a for- integer and p  1mod4 .  mula which gives the degree of each vertex in   []i Throughout this paper, p and p denote prime  n  i is derived, thus the degree of its complement as well as integers which are congruent to 1 modulo 4, while q its line graph could easily be found. and and qi denote prime integers which are congruent to 3 modulo 4. All rings in this paper are assumed to be commutative with unity. The zero divisor graph for the 2. When Is L  n[]i  Eulerian or ring of Gaussian integers modulo n is studied in [1] Planner and [2], the complement of this graph is discussed in [3]. While the line graph of the zero divisor graph for the ring If G is a connected graph. Then G is Eulerian if and

Copyright © 2012 SciRes. OJDM M. GHANEM ET AL. 25 only if every vertex of G has even degree. For a finite   [],im 3 are the union of a nullgraph and a ring R , the line graph LR of a connected graph  qm  R is Eulerian if and only if all vertices of R connected graph [3], we have the following. have the same parity ( see the proof of Lemma 3.10, [5]). Theorem 3.1 Li n[] is connected if and only if On the other hand, if G has both even and odd vertices, 2 npqqq 2, , , 12. then so is its complement. So, for a connected graph Theorem 3.2 If nm=2m , 2 or nqm=,m  3, then   []i , the graph Li  [] is Eulerian if and n n diam L  [] i =2.   n  only if all vertices in  n []i are either even or all Proof. 1) Assume that nm=2m , 2 and vertices in  n []i are all odd. But  n[]i is con- mm nected if np ,2 , qqq , 12 [3] and  n []i is x =,=,=,=xxiyyyizzziwwwi12 12  12  1  2 Eulerian if np=2, or n is a product of distinct odd primes [1]. It is easy to show that all vertices of are two nonadjacent vertices in VL  [] i . Since 2  n   n []i are odd if and only if nq= . This proves the following theorem. for every abi   [] i, a and b are both even or 2m Theorem 2.1 Li  [] is Eulerian if and only if n odd [1], we have three cases: n is a product of distinct odd primes. Case I: for i =1,2, xiii,,yz and wi are odd. Then A planar graph is a graph that can be embedded in the we have the path [,x yxzzw ]  [,]  [, ]. plane, i.e., it can be drawn on the plane in such a way Case II: for i =1,2, x or y is odd(even) and z that its edges intersect only at their endpoints. i i i or wi is even (odd). Assume that x12, x are even and Next we determine when the graph Li n [] is zz12, are odd. Then we have the path planar. [,x yxzzw ]  [,]  [, ]. In a graph G the maximum vertex degree and the Case III: for i =1,2, xiii,,yz and wi are even. minimum vertex degree will be denoted by  G and tst s   Then [,x yii ]= 2112 2 , 2 2 2 and  G , respectively.  112 2 [,zw ]= 2ts33 2i , 2ts44 2 i where  ,  The following theorem characterizes graphs G whose  3344 ii line graph LG  is planer. are odd and 1, tsii m for 14i . If tst112,,, or Theorem 2.2 [6] m A nonempty graph G has a planer line graph LG  s2 <   , say t1 , then tst334,, or s41< mt , say t3 . if and only if 2 1) G is planer. So, we have the path [,x yxzzw ] [,]  [, ]. Now 2) G 4 , and suppose that m is odd. Then 3) if deg v =4, then v is a cut vertex. m 1 G  a) If ts== ,   , for i =1 or 2, say for ii2 i i The graph  n[]i is planer if and only if n =2,5 i =1, then tst,, or s < mt , say t . Hence, we or q2 [3]. For n =2, q2 , LiK  [] = . While 334 41 3 n 0 have the path [,x yxzzw ]  [,]  [, ]. for n =5,  n[]iKK = 44, this graph is regular of degree 3. m 1 b) If ti or si = and tsii , for i =1 or 2, say Thus we obtain the following. 2  for i =1, then we have a path Theorem 2.3 The graph Lin [] is planer if and only is n =5. [,x yxzzw ]  [,]  [, ] or [,x yxwzw ]  [, ]  [, ]. 3. The Diameter of m 1 L   []i  c) If tsii== ,= i i, for i =1 or 2, say for n 2 For a connected graph G , the distance, duv, , m 1   i =1, then ts== implies that    . Other- between two vertices u and v is the minimum of the 22 2 22 lengths of all uv paths of G . The eccentricity of a m 1 wise t or s  . Then we have a path vertex v in G is the maximum distance from v to 2 2 2 any vertex in G . The diameter of G , diam G , is the [,x yyzzw ]  [,]  [, ] or maximum eccentricity among the vertices of G . Since [,x yywzw ]  [, ]  [, ]. mm m  n[]i is connected if np ,2 , qqq , 12 and each 2) Assume that nqm=, 3 and of   []i and   []i is the union of two qqiqqitsts1122,, p  qq12  112 2

ts33ts44 complete graphs, while   []i and 3344qqiqqiVLi,[] n 2m  

m diam L [] i =3. Then tst112,, or s2 < , say t1 . Hence tst334,, or   n  2 s41< mt , say t3 . Then we have the path tsts 4. The Radius and the Girth of the Graph qqiqqi1122, 112 2 L   []i ts11ts33  n  [,1133qqiqqi ].  qqiqqits33, ts44 For a connected graph G , the radius of G , rad G , is 3344 the minimum eccentricity among the vertices of G . So, Theorem 3.3 Let R be a ring that is a product of rad G  diam G . Since for any two rings R1 and R2 with at least one of them is not [,]ab V L [] i , [,]ab and [,]ai bi are non ad- ID with more than one regular element and the other has   n  more than two regular elements. Then jacent, rad L  [] i >1. Using Theorem 3.2 gives diam L R  =3.   n  Proof. Suppose that RRR=  and R is not ID, 12 1 for nm=2m , 2 or nqm=,m  3, reg R1  2 and reg R2  3 . Let zV11  R rad L  [] i =2. and uregR221 . Clearly,   n  dz1,0 , ,0 , 0,1 , 0, u = 3 in LR. So, m m 12  Theorem 4.1 If npm=, 2 or nts= where diam L R 3 . Now, let   m  1 , t is prime integer, gcd.. t , s =1 and nqq 12,

aa,,, bb , cc ,,, dd VL R , then rad L n [] i =2. 1 2  12 12 1 2       then ab  0 or ab  0 and cd  0 or cd  0 . 11 22 11 22 Proof. Since rad L n [] i >1 to show that So, we have three cases:    Case I: ab11 0 and cd11 0 . Then rad L  [] i =2 it is enough to find a vertex   n  ac11,  regR 1 implies that zaazccELR,0 , , ,0 , , . vVL [] i with eccentricity 2. If 112112      n  m 22 And a1 or cZR11 , say a1 implies that nppabm=,= , 2, then uaauccELR,0 , , ,0 , ,  dabiabixy[,],[,]2  for every 112112     where uregRc. [,x yVL ] [] i . So rad L  [] i =2. 111  n  pm    Case II: ab22 0 and cd22 0 . Then there exists m vregRac2222,  and hence Now, assume that ntsm=, 1 and

aa12,  , 0, v 2 cc 12 , , 0, v 2  EL  R . (,x ywzVL ),(,) [] i .    n  Case III: ab11 0 and cd22 0 or ab22 0 and Then we have four cases: cd11 0 . Let ab11 0 and cd22 0 . Then Case I: t =2. Then aregR11  implies that aa,,, zc VL R and 12 12    di1,1,1,0, xywz ,,,  2. zc,=, dd or dd,,, zc VL R . 12 1 2 12 12    Case II: tp = . Then And if aZR , then ac,=, bb or 11 12 12 dabiabixywz ,1 , ,1 , , , , 2 . ac,,, bb VL R and 12 12    Case III: tq= 1 and m =1. Then s  q2 and hence ac12,=, dd 1 2 or ac,,, dd VL R .  12 1 2    there exists aV  s [] i . So,

 daxywz0,1 , 1, , ,  , ,  2 . For np= m , []i  [7] and for nmmp p nnn= with gcd.. n , n =1, []iii [] []. Case IV: tqm=, 2. Then 12 12 nnn12 Moreover reg [] i =2 and reg [] i  3 for 2 m dq,1 , 1,0 , xywz ,  , ,  2 .  m  2 . An immediate consequence of Theorem 3.3 is the following. Theorem 4.2 rad L [] i =2 if and only if m  n  Theorem 3.4 Let npm=, 2 or n is a composite  npqq 2, , , 2 or qq . such that nqq 12. Then 12

Vising [8], proved that for a connected simple graph i =1,2 and either R1 or R2 is not ID, then G with n-vertices and radius 2, the upper bound of the LR is locally connected. nn  2 Proof. Suppose that R1 is not ID, zV11  R number of edges of G is . Then Golberg [9] 2 and xy,,, zw V L  R , then we have three proved that the lower bound of numbers of edges of a cases: 31n   simple connected graph G with radius 2 is . Case I: xz==0. Then 4 zxyzzwELR,1 , , ,1 , , . So we can conclude the following. 11    Theorem 4.3 For n 2, pqq , , 2 or qq , 12 Case II: yw==0. If x, zregR 1 , then Lit  [] = implies that zxyzzwELR,1 , , ,1 , , . n 11    Otherwise there exists uregRxz{,}. So, 31ttt2 11 EL [] i  . n uxyuzwELR,0 , , ,0 , , . 42  11    The girth of a graph G , g G is the length of a Case III: xy,0 or zw,0 . Assume that xy  0 , shortest cycle contained in the graph. If the graph does then z  0 implies that there exists uregRx11  {} not contain any cycles (i.e.. it’s an acyclic graph), its satisfies girth is defined to be infinity. If abca,,, is a cycle of uxyuzwELR,0 , , ,0 , , . 11    length three in G . Then [ab , ],[ bc , ],[ ca , ],[ ab , ] is a While w  0 implies that that there exists cycle of length 3 in LG. So, gLG =3 when-    vregRw  {} satisfies ever gG=3. In [3] it is proved that the girth of 22 0,vxyvzwELR , , 0, , , .  2 22    n[]i equals 3 for nqq 2, , . So, we have the following. From Theorem 5.1 and Theorem 5.2 we conclude the 2 following. Theorem 4.4 For nqq 2, , , gL n [] i =3.   Theorem 5.3 If npm=,m  1 or n is a composite integer such that nqq , then both   []i and 5. The Locally Connected Property of the 12 n  are locally connected. Li n [] Graphs   n[]i and L  n[]i

We say that a vertex v is locally connected if the 6. When Is L  n[]i  Hamiltonian? neighborhood of v , Nv  , is connected; and G is locally connected if every vertex of G is locally con- A Hamiltonian cycle is a cycle that visits each vertex nected. exactly once (except the vertex which is both the start Theorem 5.1 If RRR=,12 reg Ri  2 for and end, and so is visited twice). A graph that contains a i=1,2 and either R1 or R2 is not ID, then R is Hamiltonian cycle is called a Hamiltonian graph. The locally connected. line graph of a graph G with more than 4 vertices and Proof. Suppose that R is not ID and 1 diameter 2 is Hamiltonian [10]. But  [],im 2 is x, yV  R. Then we have two cases: 2m Case I: x =0 or y =0. If x =0, then there exists disconnected with one isolated vertex 22mm11 i zV11 R. So zabER1,1 ,   for all and the other component, call this component H , with ab,, N xy . And if y =0, then there exists   diameter 2 [3]. So, LiLH [] . Similarly,  2m   uregRx11{} such that uNxy1,0  , . There-  fore, uabER,0 ,   for every 1    []i has a connected subgraph H with diameter ab,, N  xy . So Nxy, is connected.  qm  Case II: x  0 and y  0 . Then there exist  2 and LiLH [] . Hence, the following vregRx{}, vregRy{} and qm 11 22   vzv,0 , , zV11 R such that 112 and result is obtained. m m 0,vNxy2    , . Moreover, Theorem 6.1 If nm=2 , 2 or nqm=, 3, then vzvvzvER,0 , , 0, ,  . And for every 112212   is Hamiltonian. Lin [] ts,  Z  R , vts1,0 , or 0,vtsE2 ,   R . So Nxy, is connected.  Oberly and Sumner [11] proved that every connected, Theorem 5.2 If RRR=,12 reg Ri  2 for locally connected claw free graph (i.e. it does not contain

a complete bipartite graph K1,3 ) is hamiltonian. Since   22m 2 the line graph is claw free, using Theorem 5.3, we get the    m []iq = q 1.   q  following. m Finally we find the chromatic index of Theorem 6.2 If npm=, 2 or n is a composite integer such that nqq , then Li  [] is hamil-   m [],im 2. 12 n  p  tonian. A subset D of the vertex set VG is said to be independent if no two vertices in this set are adjacent. A 7. The Chromatic Number of the Graph clique of a graph is a maximal complete subgraph. A graph G is said to be split if it’s vertex set can be L  n[]i partitioned into two subsets A and B such that A induces a clique and B is independent in G . The edge coloring of a graph G is an assignment of Lemma 7.4 [13] Let G be a split graph. If G is colors to the edges of the graph so that no two adjacent odd, then GG =   . edges have the same color. The minimum required num- Theorem 7.5 If np= 2 , then ber of colors for the edges of a given graph is called the   []ippp =2 321. chromatic index of the graph denoted by G .  n  Lemma 7.1 [12] Proof. Since []i   , it is enough to find p222pp If G has order 2s and Gs =2 1, then   . First, we’ll show that  is GG=  .  pp22  pp22 m Theorem 7.2 If nm=2 , 2, then a split graph. Let   []i =221m 3.  n AupuU=,: 2 and p }   p  

 Proof. Note that in   m []i , the induced subgraph, 2 pv , : v U 2 and p  ,  p   mm11 H , with VH=[]22 V m i i is con- Bpp= , :  and   0,0 . 2  p   21m nected, VH=2 2, [1] and Clearly, VA =  B, A induces a clique  pp22  H =[]  m i . Since the vertex 1 i is adja- 2 and B is independent. Therefore, 22 is a cent to all other vertices in H , we have  pp Hdegi=1=23   21m . Using Lemma 6.1, split graph. Moreover, 21m   m []i =2 3.  2 22  =1,deg p   pp Since  q []i is empty graph and =[]0,:2 ip p  01, p  []iqK =2 1 is edgeless with q2 1 verti-  p   q21  =2ppp32 1 ces, we consider the case  [],iq 3.  qm  is odd. From Lemma 7.4, m Theorem 7.3 If nqm=, 3, then     []ippp =2 321.   pm    22m 2   n []iq = q 1 A graph G is said to be critical if G is connected mm11 and GG =1    and for every edge e of G , Proof. Let Aq=: qi,0 q  . we have Ge\{ } <  G . The well-known Vizing’s Then A is the set of all isolated vertices in   []i .  qm  theorem states that for a simple graph G ,  GG=  or  G 1 . So the induced subgraph, H , with the vertices       Lemma 7.6 [14]  VH=[] V i A is a connected graph, If G is a critical graph, then G has at least  qm   GG    2 of vertices of maximum degree. VH= q22m  q 2. Clearly the vertex q is adjacent to Therefore, if G is a simple graph such that for every all other vertices in H and hence, vertex v of maximum degree there exists an edge vu deg q=1 q22m  q 2 . Using Lemma 7.1, such that Gdegu   2 is more than the number

of vertices with maximum degree in G , we have 22m 2  Liqq [] = 1. GG=   [13].   n  m Theorem 7.7 If npm=, 3, then 3) If npm=,m  2, then   []ippp =2 21mm 22 1. n  Lippp  [] =2 21mm 22 1.   n  Proof. Let , V  and uv,  U  . p  pm  8. The Domination Number of   []i Then the vertices of  with maximum de- n  ppmm A subset D of the vertex set VG of a graph G is gree have the form  pv, or up,  where   0  a dominating set in G if each vertex of G , not in D , and   0 and is adjacent to at least one vertex of D . The minimum  NpvV ,= cardinality of all dominating sets in G ,  G , is called  ppmm  the domination number of G . ppvm1 ,0 : 0 , In   [],im 2, the vertex 22mm11 i is an   2m  and isolated vertex while the vertex 1 i dominates all  vertices in the second component. Therefore, Nu,= p V  ppmm   2    m []i =2. The graph  21[]iqK = 1 ,  2   q    0,pupm1 : 0 ,  .   thus    []iq =2  1. In   [],i m  3 the  q2   qm  21mm 22  So, mm =2ppp   1 . And  pp vertices qqimm11  are isolated while the vertex q is adjacent to all other vertices in the vertices of  with minimum degree  ppmm  Viqqi[] mm11:, , m1 m1  qmq  have the form  p ,0 or 0,  p  where  mi1 Np,0 = upi , : 1 and 2 so    m []iq = . Since  q  Np0,mi1 = pvi , : 1 . So  qq []iK = 22 K  12 qq1211   []ippmpmp = mm1 2 .  pm    and  p []iK = pp11 K, Therefore,  []ii = [] =2.   qq12   p  []i    2   ppmm  pm  The set D =1,0,0,1    is a minimum dominating set =2ppmm11 p m mpmp  p21m p1. for . And if nnn= , where >2ppmm1  p1  ppmm 12 gcd.. n , n =1, then []iii [] []. This  12 nnn12 But the graph  has only  ppmm graph is connected and the set D =1,0,0,1  is a 21ppmm1  p vertices of maximum degree. So, minimum dominating set for []ii [] .  nn12  21mm 22  mm=2ppp  1. m  pp Theorem 8.1 1) If nq 2, , then Since []i , the result holds.  pmmmpp   []i =2. Since the edge coloring of any graph leads to a vertex  n  coloring of its line graph, we obtain the following. 2 m Corollary 7.8 1) If nm=2 , 2, then 2)    2 []iq = 1 and  q  21m  Li n [] =2 3.   2    m []iqm = , 3. m  q  2) If nqm=, 3, then 

9. The Domination Number of L   []i and not both jk,= m. The set n  mm TA= mm kj A mm is the maximum inde- The independence number of G ,  G , is the maxi- kj==  22  mum cardinality of all independent sets in G . A subset  mm  D of the edge set VG of a graph G is an edge 11  pendent set, while SA= 22  induces a dominating set in G if each edge of G , not in D , is jk=1 =1 kj adjacent to at least one edge of D . The minimum  cardinality of all edge dominating sets in G ,   G , is   maximum complete subgraph in   []i . There are called the edge domination number of G . The minimum  qm  cardinality of all independent edge dominating sets, some edges joining S to T , and   []i has no  iG , is called the independence edge domination  qm  number of G . The study of the domination number of other adjacency. Easy calculations give the line graph of G leads to the study of edge or line 2 Aqq=1 22mk  j when 1,kj m 1,  kj  domination number of G , i.e. LG=  G . On mj mj1 mk mk1 Aqmj =  q and Aqkm =  q when the other hand, for any graph G , iGG=  [15]. If S is an independent set in G , then S induces a m 2 complete graph in G . While if S induces a complete kj,  m. While Tq=12  and graph in G , then it is independent in G . Recall that 2 mm  []i  [2]. Then the sets, 2  22mm2 Sq=122 q  . AU=2:j   , jm=1,2, ,2 1 form a  jm22  j  partition for the set V   . Clearly, the set Thus we obtain the following theorem.  22m  Theorem 9.2 If nqm=,m  3, then 21m 2 TA= is the maximum independent set in mm   jm= j 2  1)   []iqq = 22  1 . n    , while the set SA= m1 induces a maxi-  2m  j=1 j 2 m m1 2)  n[]iq = 1 if m is even and q if mum complete subgraph in   . There are some m is odd. 22m Li[] = [] i edges joining S to T , no other adjacency exists in   nin  2   2m . Any edge dominating set for   2m must 3) mm  2 2 1 2  =[]=1.   iqq22  contain at least S 2 element in order to dominate n  2  S . On the other hand, this dominating set for S do-  Now, we move to the case np= m . Let minates all other edges in   . Since 22m kj 21mj AppUkj=,:  m k and U m j . Aj =2 , then S and T , could easily be com-   p   p  puted to get the following theorem. m Theorem 9.1 For nm=2 , 2. Clearly, the sets Akj where 0,kj m and not both kj,= m or 0, partition the vertices of  mm1 1)  n[]i =2 2 1 . 22mk j 2 mm and Apkj =1 p . Let  m  pp 2)  n[]i =2 1. mm1 SA10= kk A 0 Li[] = [] i  kk=1   =1   nin  3) mm  11     mm11 22  mm =[]=221. n i  SASA23=,=,kj   kj  kj=1 =1 mm  kj==  To study the graph  [],im 3, consider the  22   qm  mm  11   partition of   []i given by 22mm    qm  SASA45=and=.kj  kj jk=1 mm=1  kj kj==   AqqiU=: and   U , 22   kj q m k  q m j 

1,kj m . Note that S1 induces a complete graph in

m1 annabi  =[]:0mod x ixabi  n  . . Vertices in A are adjacent to all  ppmm k =1 k 0 abi  n m1 So,xa . bi 0 mod n  x .  0 mod . vertices except some vertices in Akm . Similarly, cdi cdi m1 k =1 vertices in A0k are adjacent to all vertices except k =1 m1 abi  n some vertices in A , and vertices in A are But Ui [] . So, x  0mod k =1 mk m0 n  cdi cdi cdi adjacent to all vertices except vertices in A0m . On the other hand A induces a complete subgraph and n 0m and hence there exists mi [] such that x = m . vertices in this set are adjacent to all other vertices except cdi those of A . Clearly S induces a complete subgraph. m0 2 Since mtcdir=     where tr,[]  i and the norm Vertices in S3 form an independent set, and are of r is less than the norm of cdi , adjacent to some vertices in SSSS A. 1245 0m ann a bi =: r r []= i [] i . By Theorem Each of S and S induces a complete subgraph and cdicdi 4 5 22 2 of [7], cdi []=icd22 =  , so the result are adjacent to some vertices in SSS123 A 0m . cd Besides, there are some edges between S4 and S5 . On holds.  the other hand, Theorem 10.2 Let vV  n[] i and mm g..cd v , n = c di. Then SAA3 =. kj mm mm  22 2 kj==  cd 1, if v 0 22  deg v =  22 2 . cd2, if v = 0 The above argument shows that The order of  []i  can be easily computed using  n Limim[] = [] i formulas given in [1]. Thus we can find the degree of  pp   each vertex in the complement of  n []i , here we 1 give the degree of each vertex in the line graph of =[]=[]  iiS  ppmm   3   []i , an analogous formula for the degree of 2  n  vertices in Li  [] could be obtained. m n 2m 1 21mm 22  =2Ppp2  2. Corollary 10.3 Let uv,[ V L  n i] ,  2 g..cd u , n = a bi and g..cd v , n = c di. Then deg[,] u v  10. The Degree of the Vertices in   []i n abcd222  2 4, if u 2  0 and v 2  0 .  222 2 2 2 and L  n[]i =5,if=0and0abcd   u v abcd222 26, if u 2 = 0 and v 2 = 0 Now, we determine the cardinality of the annihilator of  the element abi , ann a bi in  []i . This helps  n Proof. Note that, for any graph G and uv E G , find the degree of each vertex in  []i , its n deg[,]= u v deg u deg v  2.  complement, as well as the degree of each vertex in their LG()  G G corresponding line graphs. In the following we determine the degree of every

Theorem 10.1 If abi n[] i, then vertex in the graphs  n []i  when ann a bi= c22 d where g..cd a bin , = c di. nmnqm=2mm , 2, = , 3 and npm=,m  1. m Proof. Let abi n[] i and Theorem 10.4 Let nm=2 , 3 and ,  are odd. g..cd a bin , = c di. Then Then in  n []i  ,

 2k mm  21,if1<

kk npq=2, , 2 . 2) deg2= deg  2 i  r m Theorem 11.1 If n = π j where π s are  j=1 j j  2k m 21,if1<k distinct Gaussian primes and m j  1 and   mmm 2 npqm 2, , , 2, then   []i is not regular. = . n  Proof. Choose two vertices π and π such that  2k m t s 22,ifkm < ππ , then gcdn.. ,π = π  gcdn.. ,π = π . So,  2 ts  ttss the result follows.  3) deg i =1. Next, we discuss regularity of the graph Proof. 1) Note that, gcd.. n , 2ks 2 i =2 minks{,} if   Li n[] and Li n [] . Clearly, if G is re- ks k   ks or    and g..cd n , 2 2 i =2 1 i 2 gular, then LG  is also regular, so if npq=, , then if and only if ks= and  =  . Moreover the graph Li n[] is regular. On the other hand, if 2 kk m G is the complete bipartite graph K , then 22=0 i if and only if k  . rs, 2 deg[,]= u v r s  2 for all vertices in LKrs, . Thus 2) Obvious. Li qq [] is regular. While 2[]iiq [] is a 3) Note that if ,  are odd, then  12  g..cd (  i , n )=1 i.  bipartite graph with partite sets m Theorem 10.5 Let nqm=, 3, ,  are rela- Ai= 1 ,0 , 1,0 , i ,0  and tively prime with q . Then in  n []i ,

BixxVi1,:  q []  2k m   qksk1, if 1 and <   ks2  0,xxV : [ i ] . deg q q i =  .  q  2k m qks2, if  Moreover, Ni1,0= B, Ni1,1=1,0 i and  2    NA0,1 = . Thus, Theorem 10.6 Let npm=,m  1, pa= 22 b and deg1,0,1,1 i i deg 1,0,0,1  i , gcd.. , p =1. Then in  n []i ,     ks and hence, L   []i is not regular. deg  a bi a bi   2q  Theorem 11.2 If ntm=,m  2, t is a prime and 2  22ks m nq , then the graph Li  [] is not regular. ab1, if kors < and ks ,  1  n    Proof. If nm=2m , 2, then  2 mm11 m1  ks   22 m deg1,22 i i  deg 2,2 i  . If ab2, if ks ,  =.2 m mm121 nqm=, 3, then degqqi ,,  degqqi  .  22s ab1, if k = 0 And if np= m , pa= 22 b, m  2 , then  22k  ab1, if s = 0 mm  deg a bi, a bi   deg a bi,. a bimm a bi 1   11. When Is L  n[]i , L  n[]i Regular? Theorem 11.3 Let RRR= 12 where R1 and R2 are commutative rings with unity with at least one of A graph G in which all vertices have the same degree them is not ID. Then LR  is not regular. is called regular graph. Proof. Suppose that R1 is not ID and Rrii= , for Regularity of   []i was studied in [1]. However, 2 n i =1,2. Let x11VR   . If x1 =0, then we provide our own proof, since it comes as an immediate consequence of Theorem 10.2. Clearly, if Nx 12,0 = 0, aaR : 0  2 npq=2, , , then  n []i is regular. If m m y , a : y ann  x11 0, x nm=2 , 2 or nqm=, 3, then the graph

 n []i has a vertex which is adjacent to all other and aR 2 if vertices and it is not complete graph, thus  n []i  is ann x11 0, x   x 1, a : a R 2 0  , hence not regular. 2 deg x12,0 , 0,1  2 r r1 4 . And if x1  0 , Now, we show that  n []i is regular if and only if

Nx12,0 = 0, a : a R  0 12. When is L  n[]i  , L  n[]i y , a : y ann  x0, x 11 Locally H? and aR 2 if ann x110, x  , hence A simple graph G is said to be locally H if the neighborhood of each vertex in VG induces the same deg x ,0 , 0,1 r r 3 . But 121 graph H . The cartesian product GH of two graphs G and H is the graph with vertex set deg[(1,0),(0,1)] = r12 r 4 . So LR is not regular.  VG  H = VG  VH  and two vertices in So as a consequence of Theorem 11.2 and Theorem VG  H are adjacent if and only if they are equal in 11.3, we conclude the following. one coordinate and adjacent in the other. Before we proceed, we give the following lemma. Theorem 11.4 The graph Li n[] is regular if 2 and only if npqqq=, ,12. Lemma 12.1 1) If GKn=,n  3, then LG  is 2 locally Kn22K . Observe that, for nq=2, ,  n []i is the empty 2) If GK=,,2mn, mn , then LG  is locally graph.  []iN K , so the line graph Kmn11 K  .  qqqq3242 1 Proof. 1) Let [,]uv V L Kn  , then  Nuv[,]=[, uaaVK ]: {,} uv Li  3 [] is regular. While  n   q   [av , ]: a V Kn { uv , }  npp[]iK11 K each of the sets [,ua ]: a V Kn {,} uv and which is regular, so is Li  [] . n [,]:av a V Kn  {,} uv induces a copy of Kn2 and since we deal with an undirected graphs, then for a fixed  a , [,ua ] and [,va ] are adjacent. Thus the result InLidegiidegi m [ ] , 1 ,1 3 1 , 2 . 2 holds. 3) Let [,]uv V L Kmn,  , with partite sets A and  In [i ] , m > 3, deg q , qi deg q2 ,. q B and with uA , vB . Then  qm    Nuv[,]=[,]:  ubb B {}} v .   [av , ]: a A { u } And in Lim [] , 2,  pm   Each set induces a complete graph Knm11, K , res- deg a bi,, a bi deg a  bi2 a  bi . So, the graph pectively. And Nuv[,] has no other edges. Thus  Nuv[,] induces KKnm11  .   m t Lin [] is not regular for ntm=, 2, is a In order for a graph to be locally H , it should be prime and nqq 23, . regular graph. Thus for the graph Li n[] , it Theorem 11.5 Let RRR=  where R and R 2 12 1 2 suffices to check the cases npqqq=, ,12, and for are commutative rings with unity such that  3 Lin [] , we consider only the cases npq=, . VR= t, Rrii= for i =1,2. If reg Ri   2 Since  []iK = and  []iK = K, and rr12 , then LR is not regular. P pp1, 1 P pp11 Proof. Since reg R  2 , for i =1,2, there exist i Li P [] is locally KKpp22 and uregR1 and uregR1 . Therefore 11 22   LiP [] is locally K p22K . In the same manner 1,0 ,uuVLR ,0 , 0,1 , 0, . Since 12   we can show that Li  [] is locally   qq12  rr12 , KK , Li  [] is locally K K and qq2222   q2  q222 12 deg1,0 , u121 ,0 = 2 t 2 r 4 2 t  2 r  4 .  Li  [] is locally K K . =deg 0,1 , 0, u q3 qq422 2 2   Theorem 12.2 The following statements are equi- So, LR is not regular.  valent.

Theorem 11.6 The graph Li n[] is regular if  1) The graph LiLinn[] [] is regular, and only if np= or q3 . 

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