A Theory of Cartesian Product and Factorization of Circulant Graphs

Just as integers can be factored into prime numbers, there are many results on decomposition of structures throughout mathematics [1]. The standard products—Cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by Imrich and Izbicki [2] and called B-products [3]. Properties of circulant graphs are extensively studied by many authors [2–19] and products of graphs have been studied for almost 50 years now. Sabidussi [20] proved that every (nonnull) graph is the unique product of prime graphs. Broere and Hattingh [3] established that the lexicographic product of two circulant graphs is again circulant. They and Sanders and George [12] established that this is not the case with other products. Alspach and Parsons [5] introduced metacirculant graphs as a generalization of circulant graphs and characterized metacirculant graphs in terms of their automorphism groups. Sanders [11] established that any Bproduct of two circulant graphs is always a metacirculant graphwith parameters that are easily described in terms of the product graphs and also established that any metacirculant graph with the appropriate structure is isomorphic to the B-product of a pair of circulant graphs. After a graph is identified as a circulant graph, its properties can be derived easily. This paper gives a detailed study of Cartesian product and factorization of circulant graphs similar to the theory of product and factorization of natural numbers. For more details on circulant graphs, see [9, 10]. Let n be a positive integer and let R be a subset of {1, 2, . . . , [n/2]}. The circulant graph C n (R) has vertices V 1 , V 2 , . . . , V n = V 0 with V i adjacent to V i+r for each r ∈ R, subscript addition taken modulo n. When discussing circulant graphs, we will often assume, without further comment, that the vertices are the corners of a regular n-gon, labeled clockwise. Circulant graphs C 7 (1, 3) and C 8 (2, 4) are shown in Figures 1(a) and 1(b). When n/2 ∈ R, edge V i V i+n/2 is taken as a single edge while considering the degree of a vertex, but as a double edge while counting number of edges or cycles in C n (R) [3, 6– 10, 13, 14, 17, 18, 21]. We generally write C n for C n (1) and


Introduction
Just as integers can be factored into prime numbers, there are many results on decomposition of structures throughout mathematics [1].The standard products-Cartesian, lexicographic, tensor, and strong-all belong to a class of products introduced by Imrich and Izbicki [2] and called -products [3].Properties of circulant graphs are extensively studied by many authors [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19] and products of graphs have been studied for almost 50 years now.Sabidussi [20] proved that every (nonnull) graph is the unique product of prime graphs.Broere and Hattingh [3] established that the lexicographic product of two circulant graphs is again circulant.They and Sanders and George [12] established that this is not the case with other products.Alspach and Parsons [5] introduced metacirculant graphs as a generalization of circulant graphs and characterized metacirculant graphs in terms of their automorphism groups.Sanders [11] established that any product of two circulant graphs is always a metacirculant graph with parameters that are easily described in terms of the product graphs and also established that any metacirculant graph with the appropriate structure is isomorphic to the B-product of a pair of circulant graphs.After a graph is identified as a circulant graph, its properties can be derived easily.This paper gives a detailed study of Cartesian product and factorization of circulant graphs similar to the theory of product and factorization of natural numbers.For more details on circulant graphs, see [9,10].
Let ,  be positive integers,  ≥ 2 and  < /2.Then,   () consists of a collection of disjoint cycles, If  = gcd(, ), then there are  such disjoint cycles and each has length /.We say that each of these cycles has period , length /, and rotation /.
Proof.The proof is by induction on the order of .

Main Results
Throughout this section, the following notation is used.Let  ∈ N and  ⊆ N. Then  = { :  ∈ }.Let [, ] denote {,  + 1, . . ., },  ∈ Z, and let  denote disjoint union of  copies of the graph . ◻  denotes the Cartesian product of graphs  and .
In this paper, Cartesian product and factorization of connected circulant graphs is studied.Moreover  4 =  2 ◻  2 and for  ̸ = 4 and nontrivial graphs  and ,   ̸ =  ◻  and in particular,  2+1 () ̸ =  2 ◻  for any  ⊆ [1, 𝑛].In any circulant graph   () the length of a periodic cycle of period  is /gcd(, ).And 2, 3, 4, and 6 are the only numbers such that in any circulant graph   () periodic cycle of length 2, 3, 4, or 6, if it exists, occurs without rotation always.This follows from the fact that for any natural number  greater than 6, there exists at least one natural number  such that 1 <  < [/2] and gcd(, ) = 1 so that   ≅   () and rotation of the periodic cycle of period  is /gcd(, ).When  = 5,  = 2 is the required value.Thus, we call the periodic cycles of length 2, 3, 4, and 6 as the irrotational periodic cycles.
Let  2 () be a circulant graph with  ∈ .In  2 (), cycle of period  is of length 2 and irrotational.Using Remark 4, for  > 2, 3-regular connected graph  2 ◻   is a circulant of the form  2 () for some  ⊆ [1, ] if and only if each edge  ,1  ,2 acts as a cycle of period  (double edge) and equal number of vertices of  (1)   (and of  (2)   ) are on each sides of  ,1  ,2 ,  = 0, 1, 2, . . .,  − 1.This is possible only when  is odd, in which case the circulant graph is nothing but  2 (2, ).The following transformation gives the required circulant graph representation of  2 ◻   when  is odd.
Conversely, let Then using Theorem 2 the length of the periodic cycle of period 2+1 in  2(2+1) () is 2(2+1)/gcd(2(2+1), 2+1), an even number.Therefore  2 ◻  2(2+1) (2 + 1) cannot be a circulant graph (see the proof of Theorem 9) and using Lemma 14, for any connected  2+1 ().This implies that 2/gcd(2, ), the length of a periodic cycle of period  in  2 (), is odd for all  ∈  since each copy of  2 in the connected circulant graph  2 ◻  2 () acts as a periodic cycle of length 2 (see the proof of Theorem 9).This is possible only when  is even for every  ∈ .This implies that  = 2  ⊆ [1, ] and thereby   () =  2 () =  2 (2  ) = 2 ⋅   (  ) which is not connected, a contradiction.This implies  must be odd.Hence the result.So far we could find out when the cross product of  2 or  4 with another circulant is also circulant.It is interesting to know, in general, whether the cross product of any two connected circulant graphs is circulant or not.If so, when is it circulant?The following give some positive results in this direction.
Conversely, let gcd(, ) = 1.Without loss of generality, let us assume that 2 <  < .Now   ◻   is a connected 4regular graph containing  disjoint copies of   , ( disjoint copies of   ) and through all the  isomorphic images of each vertex of   , there exists a cycle of length .And for all possible values of  and ,  ⋅   ≅   () and  ⋅   ≅   () are the two edge disjoint spanning circulant subgraphs of   ◻   .
Theorem 28.Let  be a connected graph of order ,  > 2. Then  2 ◻  is circulant if and only if  ≅  or  2 ◻  where  is a connected circulant graph of odd order.
Also,  2 ◻  ≅  2 (), a circulant graph implies that 2 and  ⋅  2 are two edge-disjoint spanning subgraphs of  2 ().This implies if we remove all the  copies of circulant subgraph  2 =  2 from the circulant graph  2 () ≅  2 ◻ , then the resultant graph 2 must be a circulant graph (in any circulant graph removal or addition of one or more jump sizes, if possible, will not change the property of being circulant) which implies that  is circulant.Now  and  2 ◻  are circulants and using Theorem 18 the graph  is either of odd order or product of  2 and an odd order circulant graph.Hence the result.
Theorem 29.Let  be a connected graph of order  ≥ 2. Then  4 ◻  is circulant if and only if  is circulant of odd order.
Proof.When  is a connected circulant of odd order, then using Corollary 19,  4 ◻  is circulant.Conversely, let  4 ◻  be a connected circulant, say  4 ().In any circulant graph  4 () periodic cycles, each of length 4 occur without rotation.The spanning subgraph ⋅ 4 is also a circulant subgraph in  4 () ≅  4 ◻ .And hence if we remove all the edges of the spanning circulant subgraph  ⋅  4 from the circulant graph  4 () ≅  4 ◻ , then the resultant graph 4 must be a circulant graph which implies that  is circulant.When  and  4 ◻  are connected circulant graphs, then, using Corollary 19, the order of  is odd.Hence the result.
Theorem 30.If  and  are connected graphs and  ◻  is circulant, then  and  are circulants.
Proof.Let  and  be connected graphs of order  and , respectively, ,  ∈ N. Then the order of the graph  ◻  is .Let  ◻  be circulant.For  or  = 1 or  =  = 2, the result is true.Now let 2 ≤  ≤ .When at least one of the two graphs, say , is circulant, then  is a spanning subgraph which is also a circulant subgraph of the circulant graph  ◻ .If we remove all the edges of spanning circulant subgraph  from the circulant graph  ◻  the resultant graph  is circulant which implies that  is circulant.When both the graphs  and  are not circulant, then let   ( 1 ) be a spanned subgraph of  obtained from  by removal of minimum number of edges and   ( 2 ) be a circulant graph which is obtained from  by adding minimum number of edges in .Similarly let   ( 1 ) and   ( 2 ) be the corresponding circulant graphs obtained with respect to the graph .This implies that   (Φ) ◻   (Φ) ⊆   ( 1 ) ◻   ( 1 ) ⊂  ◻  ⊂   ( 2 ) ◻   ( 2 ) ⊆   ◻   which implies, from the construction, that  ◻  ̸ = product of a circulant graph of order  and a circulant graph of order  which implies  ◻  is not a circulant graph, a contradiction to the given condition that the graph  ◻  is circulant.This implies  and  are circulant graphs.Hence the result.where  is a prime number and  ∈ N. Given that    () is connected which implies that gcd(  ,  1 ,  2 , . . .,   ) = 1.Since  is prime, the above relation implies that there exists at least one   such that gcd(  ,   ) = 1, 1 ≤  ≤ .Now the result follows from Corollary 39.
Theorem 41 (fundamental theorem of circulant graphs).Every connected circulant graph is the unique product of prime circulant graphs (uniqueness up to isomorphism).
Remark 42.If  is a connected graph such that  ≅  1 ◻  2 ◻ ⋅ ⋅ ⋅ ◻   , then the diameter of , dia() = ∑  =1 dia(  ) [23].Thus, we can find the diameter of any given circulant graph, provided that diameters of its prime circulant graphs are known.Also the above relation helps to generate (circulant) graphs of bigger diameters.

Concluding Remarks
(1) In prime factorization of connected circulants  1 ( ) =  1 and  2 =  2 act similar to 1 and 2 among the set of all natural numbers, respectively.Thus,  1 ( ) is a unit, like 1 in number theory.(5) An interesting problem is, for a given integer , finding the number of prime (composite) circulant graphs of order either equal to  or less than or equal to . (6) One can develop theories similar to the theory of Cartesian product and factorization of circulant graphs to the other standard products of circulant graphs.

( 2 )
There exist two types of prime circulant graphs of order , one with periodic cycle(s) of length  and the other without periodic cycle of length .See Theorem 37, Corollary 39, and Remark 38.(3) The theory of factorization of circulants is similar to the theory of factorization of natural numbers and one of the very few well-known mathematical structures so vividly classified (expressed) in terms of prime factors.It can be applied in cryptography.(4) POLY315.exe is a VB program developed by us to show visually how the transformation Φ , acts on   ◻   for different values of  and , ,  ∈ N.
1 ,  2 , . ..,   } and  = { 1 ,  2 , . ..,   } are said to be Adam's isomorphic if there exists a positive integer  relatively prime to  with { 1 ,  2 , . ..,   } = { 1 ,  2 , . ..,   } *  where ⟨  ⟩ *  , the reflexive modular reduction of a sequence ⟨  ⟩, is the sequence obtained by reducing each   modulo  to yield    and then replacing all resulting terms    which are larger than /2 by  −    .The following lemmas are useful to obtain one-to-one mappings.Let  and  be two nonempty sets.Let  :  →  be a mapping.Then,  is one-to-one if and only if /  is oneto-one for every nonempty subset   of .Let  and  be nonempty sets and let  1 ,  2 , . . .,   be a partition of  (each   being nonempty),  = 1, 2, . . ., .Let  :  →  be a mapping.Then  is one-to-one if and only if /  is one-to-one for every ,  = 1, 2, . . ., .
1, [/2]], ,  ∈ N; (iv) in any circulant graph   () removal or addition of one or more jump sizes, if possible, will not change the property of being circulant; (v) if  and  are nontrivial graphs of order  and , respectively, ,  ≥ 2, then  ◻  contains  number of disjoint copies of  and  number of disjoint copies of .Let  > 2. Then  2 ◻   is circulant if and only if  is odd.Furthermore, in that case,  2 ◻   ≅  2 (2, ).