A Study on Isomorphic Properties of Circulant Graphs V

Advances in Science, Technology and Engineering Systems Journal ASTES Journal Vol. 2, No. 6, 236-241 (2017) ISSN: 2415-6698 www.astesj.com

A Study on Isomorphic Properties of Circulant Graphs V. Vilfred Kamalappan * Department of Mathematics, Central University of Kerala Periye, Kasaragod, Kerala, 671316 India

ARTICLEINFO ABSTRACT Article history: C (R) denotes circulant graph C (r ,r ,...,r ) of order n for a set R = n n 1 2 h ki Received: 13 October, 2017 n r1,r2,...,rk where 1 r1 < r2 < ... < rk 2 . Circulant graph Cn(R) Accepted: 06 November, 2017 { } ≤ ≤ is said to have the Cayley Isomorphism (CI) property if whenever Cn(S) Online: 20 December, 2017 Z is isomorphic to Cn(R), there is some a n∗ for which S = aR. In this Keywords: paper, isomorphic properties of circulant∈ graphs that includes (i) Self-Complementary circulant Self-complementary circulant graphs; (ii) Type-2 isomorphism, a new graphs type of isomorphism other than already known Adam’s isomorphism of Type-1 and Type-2 isomorphic circulant graphs and (iii) Cartesian product and factorization of circulant graphs circulant graphs similar to the theory of product and factorization of Cartesian product and factor- natural numbers are studied. New abelian groups are obtained from ization these isomorphic circulant graphs. Type-2 isomorphic circulant graphs prime and composite circulant have the property that they are isomorphic graphs without Cayley graphs Isomorphism (CI) property and thereby new families. Factorization theorem, Funda- mental theorem

1 Introduction C (R) denotes circulant graph C (r ,r ,...,r ) where n h i n 1 2 k 1 r < r < ... < r n . Only connected circu- ≤ 1 2 k ≤ 2 This paper is an extension of work originally pre- lant graphs of finite order are considered, V (Cn(R)) sented in ICMSAO2017 [1] and covers the au- = v0,v1,v2,...,vn 1 with vi adjacent to vi+r for each thor’s study on a few isomorphic properties of cir- r { R, subscript addition− } taken modulo n and all cy- culant graphs that includes (i) Existence of self- cles∈ have length at least 3, unless otherwise specified, n complementary circulant graphs; (ii) Type-2 isomor- 0 i n 1. However when R, edge v v n is taken 2 i i+ 2 phism, a new type of isomorphism other than al- as≤ a single≤ − edge for considering∈ the degree of the ver- ready known Adam’s isomorphism of circulant graphs tex v or v n and as a double edge while counting the i i+ 2 that helps to obtain graphs without CI-property and number of edges or cycles in C (R), 0 i n 1. Gen- n ≤ ≤j k− abelian groups and (iii) Cartesian product and fac- erally, write C for C (1) and C (1,2,..., n ) for K . torization of circulant graphs similar to the theory of n n n 2 n We will often assume, with-out further comment, that product and factorization of natural numbers. the vertices are the corners of a regular n gon, labeled Beauty comes out of symmetry as well as asym- − clockwise. Circulant graphs C16(1,2,7), C16(2,3,5), metry. Investigation of symmetries/asymmetries of C27(1,3,8,10),C27(3,4,5,13) and C27(2,3,7,11) are structures yield powerful results in Mathematics. Cir- shown in Figures 1 - 5. Now, let us consider the fol- culant graphs form a class of highly symmetric math- lowing definitions and results that are useful in the ematical (graphical) structures. In 1846 Catalan (cf. subsequent sections. [2]) introduced circulant matrices and properties of circulant graphs have been investigated by many au- Definition 1.1. [17] Let n and r be positive integers n thors [1-20]. An excellent account of circulant matri- with n 4 and r < . Then, clearly, Cn(r) consists of a ≥ 2 ces can be found in the book by Davis [2] and circulant collection of cycles (v0vr v2r ...v0), (v1v1+r v1+2r ...v1),..., graphs in the article [11]. (vr 1v2r 1v3r 1 ...vr 1). If d = gcd(n,r), then there are d − − − − n such disjoint cycles, each of length d . We say that each of If a graph G is circulant, then its adjacency matrix n r A(G) is circulant. It follows that if the first row of the these cycles is of period r, length d and rotation d . n adjacency matrix of a circulant graph is [a1,a2,...,an], If r = , then obviously C (r) is just a 1 f actor. 2 n − then a1 = 0 and ai = an i+2, 2 i n [15, 17]. Since C (R) is just the union of the cycles C (r) for − ≤ ≤ n n Through-out this paper, for a set R = r ,r ,...,r , r R, we have a decomposition of C (R). { 1 2 k} ∈ n *Corresponding Author: V. Vilfred Kamalappan Email: [email protected], Contact No. 0091 9443577502 www.astesj.com 236 https://dx.doi.org/10.25046/aj020628 V. Vilfred / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 6, 236-241 (2017)

circulant whereas graphs given in Figures 8 and 9 are self-complementary circulant graphs.

Theorem 1.2. [17] Let r R. Then, in Cn(R), the ∈ n length of a cycle of period r is gcd(n,r) and the number of disjoint periodic cycles of period r is gcd(n,r). Theorem 2.1. [17] ( ) ( ) Corollary 1.3. [17] In C (R), the length of a cycle of If Cn R Cn S , then there is a n ( ) = period r is n if and only if gcd(n,r) = 1, r R. bijection f from R to S so that for all r R, gcd n,r ∈ gcd(n,f (r)). ∈ Remark 1.4. [17] Let R = k. Then, the circulant | | Proof. The proof is by induction on the order of graph Cn(R) for a set R = r1,r2,...,rk is (2k 1) regular R. n { } − − if 2 R and 2k regular otherwise. ∈ − Theorem 2.2. [9] If graph G of order n is self- The following Lemmas are useful to obtain one-to- complementary, then n 0,1 (mod 4). one mappings. ≡ Theorem 2.3. [17] If Cn(R) for a set R = r1,r2,...,rk Lemma 1.5. [15] Let A and B be two non-empty sets. is self-complementary, then n 1 (mod 4). { } Let f : A B be a mapping. Then, f is one-to-one if and ≡ → Proof. Using Theorem 2.2, we get n 0,1 (mod 4). only if f / is one-to-one for every non-empty subset A0 ≡ A0 When n is even, a circulant graph is of even degree of A. if and only if its complement is of odd degree since Lemma 1.6. [15] Let A and B be non-empty sets and any circulant graph is a regular graph. Thus, self- complementary circulant graph of even order doesn’t A1,A2,...,Ak be a partition of A (each Ai being non- empty, i = 1,2,...,k). Let f : A B be a mapping. Then exist. Hence, we get the result. f is one-to-one if and only if f /→ is one-to-one for every Ai Theorem 2.4. [17] ( ) = If Cn R for a set R r1, r2, ..., rk i, i = 1,2,...,k. is self-complementary, then n = 4m + 1, k ={m and S is} h i| i| even where S = j : gcd(n,j) = gcd(n,i), 1 j n , for i { ≤ ≤ 2 } h n i 2 On self-complementary circu- all i, i = 1,2,..., 2 . lant graphs Proof. Using Theorem 2.3, we get, n = 4m + 1. This implies that the degree (of each vertex) of a self- In 1962, Horst Sachs [14] proved that the sufficient complementary circulant graph of order 4m + 1 is 2m condition for the existence of a self-complementary which implies k = m. circulant graph of order n is that every prime factor Let C4m+1(R) be a self-complementary circulant p of n should satisfy p 1 (mod 4). He also conjec- ≡ graph for R = r1, r2, ..., rm . If it contains a cy- tured that the sufficient condition is a necessary one. cle of period r {(and of length} n , using Theo- We have proved that the self-complementary circu- gcd(n,r) rem 1.2), then its complement also contains a peri- lant graph on n vertices does not exist if n has any odic cycle of period, say, s such that n = n , prime factor which is not of the form 4m + 1, m N. gcd(n,s) gcd(n,r) ∈ h n i Thereby, we establish that the sufficient condition is 1 r,s , using Theorem 1.2. This implies that ≤ ≤ 2 also a necessary one. The proof is based on counting gcd(n,s) = gcd(n,r). Here we consider that the cycles the number of disjoint cycles of a particular length in of periods r and n r are the same in C (r ,r ,...,r ), i − i n 1 2 k Kn and is given in this section [4, 17]. Graphs given 1 i k. Combining the above arguments, we get the in Figures 6 and 7 are self-complementary but not of result.≤ ≤ www.astesj.com 237 V. Vilfred / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 6, 236-241 (2017)

Remark 2.5. The above theorem states that if a self- graph of order n is that each prime factor p of n should complementary circulant graph of order n exists, then satisfy p 1(mod 4). n = 4m + 1 and so it is 2m-regular and the number of ≡ periodic cycles of length i in K is always even for each i, h i n n 3 On Isomorphism of Circulant 1 i 2 . ≤ ≤ Graphs Theorem 2.6. [17] Self-complementary circulant graph of order n doesn’t exist when n has any prime factor of the In this section, Type-2 isomorphism, a new type of iso- form 2(2m 1) + 1, m N. − ∈ morphism different from already known Adam’s iso- Proof. Using Theorem 2.3, n = 4a + 1, a + 1 N. morphism of circulant graphs, main results related to n ∈ Let n = pn1 pn2 ...p j where p ,p ,...,p are the it and families of new abelian groups obtained from 1 2 j 1 2 j isomorphic circulant graphs are presented. Type-2 (odd) prime factors of n. Let p 4m + 1 for at least i , isomorphic circulant graphs have the property that one i, 1 i j and for any m N. This implies, they are isomorphic graphs without Cayley Isomor- p = 2(2q≤ 1)≤ + 1 for some q N∈. Consider the cir- i − ∈ phism (CI) property. culant graph C4a+1(1,2,...,2a) K4a+1 = Kn. Let r be the natural number such that r = n . pi Definition 3.1. [12] A circulant graph Cn(R) is said Aim To find out all natural numbers lying between 1 to have the CI-property if whenever Cn(S) is isomorphic Z and n such that g.c.d. of each one of them with n is to Cn(R), there is some a n∗ for which S = aR. exactly r. ∈ Z Let gcd(n,pr + s) = r where p,s + 1 N such that Lemma 3.2. [16] Let S be a non-empty subset of n ∈ and x Z . Define a mapping Φ : S Z such that 0 s < r and 1 pr + s n = rpi. This implies that n n,x n ≤ ≤ ≤ Φ (s)∈= xs for every s S under multiplication→ mod- s = 0 and so gcd(n,rp + s) = gcd(n,rp) = gcd(rpi,rp) = r n,x ulo n. Then, Φ is bijective∈ if and only if S = Z and where 1 pr pir = n. Thus we get gcd(p,pi) = 1 n,x n ≤ ≤ gcd(n,x) = 1. where p pi which is greater than 1. Therefore the possible values≤ of p are 1,2,...,p 1. i Definition 3.3. [3] Circulant graphs C (R) and C (S) Thus r,2r,3r,...,(p 1)r are the− only numbers ly- n n i for R = r ,r , ...,r and S = s ,s ,...,s are Adam’s ing between 1 and n such− that g.c.d. of n with each one 1 2 k 1 2 k isomorphic{ if there exists} a positive{ integer} x relatively of them is exactly r. This implies r,2r,3r,...,(p 1)r i prime to n with S = xr , xr , ..., xr where < r > , the are the possible periods of cycles of length p each,− in 1 2 k n∗ i n∗ i reflexive modular reduction{ of a sequence} < r > is the se- h n i i Cn(1,2,..., ) since the length of a cycle of period rp 2 quence obtained by reducing each ri modulo n to yield ri0 h n i n rpi pi in Cn(1,2,..., ) is = = = pi and then replacing all resulting terms ri0 which are larger 2 gcd(n,rp) gcd(rpi ,rp) gcd(pi ,p) n for p = 1,2,...,p 1, using Theorem 1.2. than 2 by n ri0. i h− i − In C (1,2,..., n ), Z n 2 Lemma 3.4. [16] Let m,r,t n gcd(n,r) = m > 1 n ∈ ∈ the cycles of period r and n r(= (pi 1)r) are the and 0 t 1. Then the mapping Θ : Z Z − − m n,r,t n n same, defined≤ by ≤Θ −(x) = x+jtm for every x Z under→ arith- n,r,t ∈ n the cycles of period 2r and (pi 2)r are the same, metic modulo n is bijective where x = j+qm, 0 j m 1, ... − n Z ≤ ≤ − 0 q m 1 and j,q n. (pi 1)r (pi +1)r ≤ ≤ − ∈ the cycles of period −2 and 2 are the same. pi 1 Theorem 3.5. [16] Let V (Cn(R)) = v0,v1,...,vn 1 , Thus, there are 2− number of possible distinct { − } pi 1 V (Kn) = u0, u1,...,un 1 , r R and gcd(n,r) = m > periodic cycles of (periods r,2r,..., − and) length pi, { − } ∈ h i 2 1. Then the mapping Θn,r,t : V (Cn(R)) V (Kn) de- n → each in Cn(1,2,..., 2 ). fined by Θn,r,t(vx) = ux+jtm and Θn,r,t((vx, vx+s)) = pi 1 Z Now, − = 2q 1 is an odd number. This im- (Θn,r,t(vx),Θn,r,t(vx+s)) for every x n, x = j + qm, 2 n ∈ plies, any circulant− graph of order and its comple- 0 j m 1, 0 q,t 1 and s R, under sub- n ≤ ≤ − ≤ ≤ m − ∈ mentary circulant graph contain unequal number of script arithmetic modulo n, for a set R = r1, r2, ..., rk, { periodic cycles of length p , each. This implies that n rk, n rk 1, ..., n r1 is one-to-one, preserves adja- i − − − − } n self-complementary circulant graph of order does cency and Θn,r,t(Cn(R)) Cn(R) for t = 0,1,2,..., 1. n m − n’t exist when n contains any prime factor of the form 2(2q 1) + 1, q N, by Remark 2.5. − ∈ Definition 3.6. [16] For a given Cn(R) and for a par- n Thus, when n = 9,21,33,49,57,69,77,81,93, etc., ticular value of t, 0 t m 1, if Θn,r,t(Cn(R)) = Cn(S) n ≤ ≤ − self-complementary circulant graph doesn’t exist on n for some S [1, ] and S , xR for all x Φn under re- ⊆ 2 ∈ vertices, by Theorem 2.6, even though in each case, flexive modulo n, then Cn(R) and Cn(S) are called Type-2 n 1 (mod 4). isomorphic circulant graphs w.r.t. r, r R. In this case, ≡ Z ∈ Now, by combining Theorem 2.6 and the sufficient subsets R and S of n are called Type-2 isomorphic sub- Z condition for the existence of a self-complementary sets of n w.r.t. r. circulant graph of order n, we get the following result. Clearly, Type-2 isomorphic circulant graphs are Theorem 2.7. [17] The necessary and sufficient condi- circulant graphs without CI-property. We obtained tion for the existence of a self-complementary circulant the following results on Type-2 isomorphism. www.astesj.com 238 V. Vilfred / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 6, 236-241 (2017)

Theorem 3.7. [16] For n 2, k 3, 1 2s 1 2n 1, Theorem 3.12. [1] Let p be an odd prime and k 3. ≥ ≥ ≤ − ≤ − ≥ n , 2s 1,R = 2s 1,4n 2s +1,2p1,2p2,...,2pk 2 and Then, for i = 1 to p, di = (i 1)np + 1 and Ri = di, − { − − − } 2 2 2 −2 2 {2 S = 2n (2s 1), 2n + 2s 1,2p1,2p2,...,2pk 2 , cir- np di,np + di,2np di,2np + di,3np di, 3np + { − − − − } − 2 − 2 3 − culant graphs C8n(R) and C8n(S) are Type-2 isomorphic di,...,(p 1)np di,(p 1)np +di,np di,pp1,pp2,..., − 3 − − 3 − 3 (and without CI property) where gcd(p1,p2,...,pk 2) ppk 2, p(np pk 2), p(np pk 3), ..., p(np p1) , − N − − − − − − − } = 1 and n,s,p1,p2,...,pk 2 . circulant graphs Cnp3 (Ri) are Type-2 isomorphic (and − ∈ without CI-property) where gcd(p1,p2,...,pk 2) = 1 and Theorem 3.8. [16] For R = 2r 1,2s 1,2p1,2p2,..., N − n{ − − n n,p1,p2,...,pk 2 . 2pk 2 , n 2, k 3, 1 t [ 2 ], 1 2r 1 < 2s 1 [ 2 ], − ∈ − } ≥ ≥ ≤ ≤ ≤ − − ≤ N gcd(p1,p2,...,pk 2) = 1 and n,r,s,t,p1,p2,...,pk 2 , Theorem 3.13. [1] Let p be an odd prime, k 3, − − ∈ 2 ≥ if Θn,2,t(Cn(R)) and Cn(R) are Type-2 isomorphic circu- 1 i p, di = (i 1)np + 1,Ri = di, np di, ≤2 ≤ 2 − 2 2 { 2 − lant graphs for some t, then n 0 (mod 8), 2r 1 + 2s 1 np + di, 2np di, 2np + di, 3np di, 3np + di, n n 3n n ≡ n − − 2 − 2 −3 = 2 , t = 8 or 8 , 2r 1 , 8 , 1 2r 1 4 and n 16. ..., (p 1)np di, (p 1)np + di, np di, pp1, pp2, − ≤ − ≤ ≥ − −3 − 3 − 3 ..., ppk 2, p(np pk 2), p(np pk 3), ..., p(np p1) , Definition 3.9. [1] Let Adn = Φn,x : x Φn , − − − − − − } { ∈ } T 2(Ri) = Θnp3,p,jn(Ri): j = 1,2,...,p ,T 2(Cnp3 (Ri)) = Adn(R) = Φn,x(R): x Φn = xR : x Φn and { } { ∈ } { ∈ } Θnp3,p,jn(Cnp3 (Ri)) : j = 1,2,...,p , gcd(p1,p2,...,pk 2) Adn(Cn(R)) = T 1n(Cn(R)) = Φn,x(Cn(R)) : x Φn = { N} − { ∈ } = 1 and n,p1,p2,...,pk 2 . Then T 2np3,p(Ri) Cn(xR): x Φn for a set R = r1,r2,...,rk, n rk, n − ∈ { ∈ Z} { − − = T 2(Rj ),T 2np3,p(Cnp3 (Ri)) = T 2(Cnp3 (Rj )) and rk 1,...,n r1 n. Define 0 0 in Adn(Cn(R)) such that − − } ⊆ ◦ (Vnp3,p(Ri), ), (Vnp3,p(Cnp3 (Ri)), ) and (T 2np3,p(Ri), ) Φn,x(Cn(R)) Φn,y(Cn(R)) = Φn,xy(Cn(R)) and Cn(xR) ◦ ◦ ◦ ◦ are abelian groups, 1 i,j p. Moreover, (T 2np3,p(Cnp3 ( Cn(yR) = Cn((xy)R) for every x,y Φn, under arith- ≤ ≤ R )), ) is the Type-2 group of order p on C 3 (R ) w.r.t. metic◦ modulo n. Clearly, Ad (C (R))∈is the set of all cir- i np i n n r = p,◦1 i,j p. culant graphs that are Adam’s isomorphic to Cn(R) and ≤ ≤ (Ad (C (R)), ) = (T 1 (C (R)), ) is an abelian group Circulant graphs C (1,2,7) and C (2,3,5) are n n ◦ n n ◦ 16 16 and we call it as the Adam’s group or Type-1 group on Type-2 isomorphic and C27(1,3,8,10),C27(3,4,5,13) C (R) under . and C (2,3,7,11) are also. See Figures 1–5. n 0◦0 27 Definition 3.10. [1] Let V (Cn(R)) = v0,v1,...,vn 1 , { −Z} V (Kn) = u0, u1, ..., un 1 , r R, m,q,t,t0,x n 4 Cartesian Product and Factoriza- such that {gcd(n,r) = m− >}1, x∈ = j + qm, 0 ∈ j n ≤Z ≤ tion of Circulant Graphs m 1 and 0 q,t,t0 m 1. Define Θn,r,t : n Z −and Θ ≤: V (C (R≤)) −V (K ) such that Θ (→x) n n,r,t n n n,r,t Just as integers can be factored into prime num- = x + jtm, Θ (v ) = u→ and Θ ((v ,v )) = n,r,t x x+jtm n,r,t x x+s bers, there are many results on decompositions of (Θ (v ),Θ (v )) for every x Z and s R, under n,r,t x n,r,t x+s n structures throughout mathematics [6]. The standard subscript arithmetic modulo n. Let∈s Z ,V ∈= Θ : n n,r n,r,t products - Cartesian, lexicographic, tensor, and strong t = 0,1,..., n 1 ,V (s) = Θ (s):∈ t = 0,1,...,{ n 1 m n,r n,r,t m - all belong to a class of products introduced by Im- and V (C (R−)) }= Θ (C{ (R)) : t = 0,1,..., n −1 }. n,r n n,r,t n m rich and Izbicki and called [8]. In this Define in V such{ that Θ Θ = Θ − }, B products 0 0 n,r n,r,t n,r,t0 n,r,t+t0 section, a few important results− that are obtained in (Θ ◦Θ )(x) (= Θ (Θ (x◦)) = Θ (x + jt m) n,r,t n,r,t0 n,r,t n,r,t0 n,r,t 0 our study of Cartesian product and factorization of = (x + ◦jt m) + jtm = x + j(t + t )m) = Θ (x) and 0 0 n,r,t+t0 circulant graphs, similar to the theory of product and Θ (C (R)) Θ (C (R)) = Θ (C (R)) for every n,r,t n n,r,t0 n n,r,t+t0 n factorization of natural numbers, are presented (For Θ ,Θ ◦V where t + t is calculated under addi- n,r,t n,r,t0 n,r 0 details see [15]). Graphs and (5 6) are iso- tion modulo ∈n . Clearly, (V (s), ) and (V (C (R)), ) C5 C6 C30 , m n,r n,r n morphic and are given in Figures 12 and 13. One can are abelian groups for every s Z ◦. ◦ ∈ n see the difficulty of this study from this example. Vn,r (Cn(R)) contains all isomorphic circulant Definition 4.1. [9] The cross product or Cartesian graphs of Type-2 of Cn(R) w.r.t. r, if exist. Let product of two simple graphs G(V,E) and H(W,F) is the T 2n,r (Cn(R)) = Cn(R) Cn(S): Cn(S) is Type-2 iso- { } ∪ { simple graph G H with vertex set V W in which morphic to Cn(R) w.r.t. r . Thus, T 2n,r (Cn(R)) = × } two vertices u = (u1,u2) and v = (v1,v2) are adjacent if Cn(R) Θn,r,t( Cn(R)) : Θn,r,t(Cn(R)) = Cn(S) and { } ∪ { and only if either u1 = v1 and u2v2 F or u2 = v2 and Cn(S) is Type-2 isomorphic to Cn(R) w.r.t. r, 0 t ∈ n 1 = Θ (C (R)) Θ (C (R)) : Θ (C≤(R≤)) u1v1 E. m − } { n,r,0 n } ∪ { n,r,t n n,r,t n ∈ = Cn(S) and Cn(S) is Type-2 isomorphic to Cn(R), Theorem 4.2. [15] Let G be a connected graph of order 0 t n 1 V (C (R)) and (T 2 (C (R)), ) is ≤ ≤ m − } ⊆ n,r n n,r n ◦ n, n > 2. Then, P2 G is circulant if and only if G H a subgroup of (V (C (R)), ). Clearly, T 1 (C (R)) n,r n ◦ n n ∩ or P2 H where H is a connected circulant graph of odd T 2 (C (R)) = C (R) . And C (R) has Type-2 isomor- n,r n { n } n order. phic circulant graph w.r.t. r if and only if T 2n,r (Cn(R)) , Cn(R) if and only if T 2n,r (Cn(R)) Cn(R) , Φ if Theorem 4.3. [15] Let G be a connected graph of or- { } ∩ { } and only if T 2n,r (Cn(R)) > 1 [1]. der n 2. Then C4 G is circulant if and only if G is | | circulant≥ of odd order. Definition 3.11. [1] For any circulant graph Cn(R), if T 2n,r (Cn(R)) , Cn(R) , then (T 2n,r (Cn(R)), ) is called Theorem 4.4. [15] If G and H are connected graphs the Type-2 group{ of C (}R) w.r.t. r under ‘ . ◦ and G H is circulant, then G and H are circulants. n ◦0 www.astesj.com 239 V. Vilfred / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 6, 236-241 (2017)

Graphs P2C3 and P2C4 are given in Figures 10 (ii) G C2m+1(R); H C2n+1(S),P2 C2n+1(S),C4 k and 11 and C5 C6 and C30(5,6) C5 C6 in 12 and C2n+1(S) or C2k(2n+1)(S) and gcd(2m+1,2 (2n+1)) 13, respectively. = 1, k N. ∈ (iii) G P2 C2m+1(R); H C2n+1(S) or P2 C2n+1(S) and gcd(2m + 1,2n + 1) = 1.

(iv) G C k (R) P C k 1 (T ) for any 2 (2m+1) , 2 2 − (2m+1) k C k 1 (T ); H C (S) and gcd(2 (2m + 2 − (2m+1) 2n+1 1),2n + 1) = 1, k N. ∈ (v) G C4 C2m+1(R); H C2n+1(S) and gcd(2m + 1,2n + 1) = 1.

(vi) G C k (R) C C k 2 (T ),P 2 (2m+1) , 4 2 − (2m+1 2 C k 1 (U) for any C k 2 (T ) and 2 − (2m+1) 2 − (2m+1 k C k 1 (U); H C (S) and gcd(2 (2m + 2 − (2m+1) 2n+1 1),2n + 1) = 1, k N, k 2. ∈ ≥ Deﬁnition 4.6. [15] A non-trivial graph G is said to be prime if G = G1G2 implies G1 or G2 is trivial; G is composite if it is not prime.

Deﬁnition 4.7. [15] If Cm(R),Cn(S) and Cmn(T ) are circulant graphs such that Cm(R) Cn(S) Cmn(T ), then we say that Cm(R) and Cn(S) are divisors or factors of Cmn(T ). Thus for any connected circulant graph, the graph and C1( ) = K1 are always divisors and so we call them as improper divisors of the circulant graph. Divisors which are integer multiple of improper divisors also be called as improper divisors of the circulant graph. This doesn’t arise since we consider divisors of connected graphs only. Divisor(s) other than improper divisors is called proper divisor(s) of the circulant graph. Deﬁnition 4.8. [15] A circulant graph whose only divisors are improper is called a prime circulant graph. Other circulant graphs are called composite circulant graphs. Theorem 4.9. [15] [Factorization Theorem On Circu- lant Graphs] m Let m and n be relatively prime integers. If R [1, 2 ], S [1, n ] and T [1, mn ] with T = dnR dmS⊆for some d ⊆ 2 ⊆ 2 ∪ such that gcd(mn,d) = 1, then Cmn(T ) Cm(R) Cn(S).

Theorem 4.10. [15] If n , 4 and 1 R, then Cn(R) is a prime circulant. ∈ Corollary 4.11. [15] If n , 4 and R contains an integer relatively prime to n, then Cn(R) is prime circulant. Corollary 4.12. [15] If n is a prime power other than 4 and Cn(R) is connected, then Cn(R) is prime circulant for all R , φ. Theorem 4.13. [15] [Fundamental Theorem of Circu- lant Graphs] Every connected circulant graph is the unique product of Theorem 4.5. [15] Let G and H be connected graphs, prime circulant graphs (uniqueness up to isomorphism). each of order > 2. Then G H is circulant if and only if G and H are circulants and satisfy one of the following conditions: Remark 4.14. [13, 15] If G is a connected graph such that G G1 G2 ... Gk, then the diameter of G, Pk (i) G Cm(R); H Cn(S) and gcd(m,n) = 1. dia(G) = i=1 dia(Gi). www.astesj.com 240 V. Vilfred / Advances in Science, Technology and Engineering Systems Journal Vol. 2, No. 6, 236-241 (2017)

Thus, we can find the diameter of any given circu- Kanyakumari District, Tamil Nadu, India for provid- lant graph, provided diameters of its prime circulant ing facilities and CU Kerala and Lerroy Wilson Foun- graphs are known. Also the above relation helps to dation, India (www.WillFoundation.co.in) for provid- generate (circulant) graphs of bigger diameters. ing financial assistance to do this research work. Remark 4.15. [15] References 1. In prime factorization of connected circulants C1() 1. V. Vilfred, “A few properties of circulant graphs: Self- = K1 and C2 = P2 act similar to 1 and 2 among the complementary, isomorphism, Cartesian product and factor- th set of all natural numbers, respectively. Thus, C1() ization” in 7 International Conference on Modeling, Simu- is a unit, like 1 in number theory. lation and Applied Optimization (ICMSAO2017), American University of Sharjah, Sharjah, April 2017. 2. There exist two types of prime circulant graphs of 2. P. J. Davis, Circulant Matrices, Wiley, New York, 1979. order n, one with periodic cycle(s) of length n and 3. A. Adam, “Research problem 2-10”, J. Combinatorial The- the other without periodic cycle of length n. ory, 3, 393, 1967. 3. The theory of factorization of circulants is similar 4. B. Alspach, J. Morris and V. Vilfred, “Self-complementary to the theory of factorization of natural numbers circulant graphs”, Ars Com., 53, 187–191, 1999. and one of the very few well-known mathematical 5. B. Alspach and T. B. Parsons, “Isomorphism of circulant structures so vividly classified (expressed) in terms graphs and digraphs”, Discrete Math., 25, 97–108, 1979. of prime factors. It can be applied in cryptography. 6. R. Arratia, A. D. Barbour and Simon Tavare, “Ran- dom Combinatorial Structures and Prime Factorizations”, 4. We developed VB programs POLY215.exe and A.M.S.Notices, 44, 903–910, 1997. POLY315.exe to show visually how the transforma- 7. F. T. Boesch and R. Tindell, “Circulant and their connectivi- ties”, J. Graph Theory, 8, 487–499, 1984. tions Θn,r,t and Φm,n act on Cn and Cm Cn, respectively for different values of m and n, m,n N. 8. B. Elspas and J. Turner, “Graphs with circulant adjacency ∈ matrices”, J. Combinatorial Theory, 9, 297–307, 1970. 5. An interesting problem is, for a given integer n, 9. F. Harary, Graph Theory, Addison-Wesley, Reading, M.A., finding the number of prime (composite) circulant 1969. graphs of order either equal to n or less than or 10. W. Imrich and H. Izbicki, “Associative products of graphs”, equal to n. Monatsh. Math., 80, 277–281, 1975. 6. One can develop theories similar to the theory of 11. I. Kra and S. Simanca, “On Circulant Matrices”, AMS No- Cartesian product and factorization of circulant tices, 59 (3), 368–377, 2012. graphs to the other standard products of circulant 12. C. H. Li, “On isomorphisms of finite Cayley graphs-a sur- graphs. vey”, Discrete Math. 256, 301–334, 2002. 13. G. Sabidussi, “Graph multiplication”, Math. Z., 72, 446– 457, 1960. Conclusion This study covers a few isomorphic properties of circulant graphs. One can go for a simi- 14. H. Sachs, “Uber selbstkomplementare Graphen”, Publ. Math. Debrecen, 9, 270–288, 1962. lar study on Cayley graphs. 15. V. Vilfred, “A Theory of Cartesian Product and Factoriza- tion of Circulant Graphs”, Hindawi Pub. Corp. - J. Discrete Acknowledgment The author expresses his sincere Math., Vol. 2013, Article ID 163740, 10 pages. thanks to Prof. Lowell W Beineke, Indiana-Purdue 16. V. Vilfred, “New Abelian Groups from Isomorphism of Cir- University, U.S.A., Prof. Brian Alspach, University culant Graphs”, Proce. of Inter. Conf. on Applied Math. and of Newcastle, Australia, Prof. M.I. Jinnah, Univer- Theoretical Computer Science, St.Xaviers Catholic Engineer- sity of Kerala, Trivandrum, India and Prof. V. Mohan, ing College, Nagercoil, Tamil Nadu, India, xiii–xvi, 2013. P Thiyagarayar College of Engineering, Madurai, Tamil 17. V. Vilfred, “ -labelled Graphs and Circulant Graphs”, Nadu, India for their valuable suggestions and guid- Ph.D. Thesis, University of Kerala, Trivandrum, India, 1994. ance and Dr. A. Christopher, Dr. P. Wilson and Mr. R. 18. V. Vilfred and P. Wilson, “New Families of Circulant graphs Satheesh of S.T. Hindu College, Nagercoil, India for without Cayley Isomorphism Property with mi = 5”, Int. Journal of Scientific and Innovative Mathematical Research, their assistance to develop the VB programs to show 3 (6), 39–47, 2015. how the transformations defined on circulant graphs 19. V. Vilfred and P. Wilson, “New Family of Circulant Graphs to obtain their isomorphic graphs is taking place. He without Cayley Isomorphism Property with mi = 7”, IOSR also expresses his gratitude to Central University of Journal of Mathematics, 12, 32–37, 2016. Kerala (CU Kerala), Periye - 671 316, Kasaragod, Ker- 20. H. Zhang, “Self-complementary symmetric graphs”, J. ala, India and St. Jude’s College, Thoothoor - 629 176, Graph Theory, 16, 1–5, 1992.