The Existence of Selfcomplementary Circulant Graphs (original) (raw)

Almost self-complementary circulant graphs

Discrete Mathematics, 2004

An almost self-complementary graph is a graph isomorphic to its complement minus a 1-factor. An almost self-complementary circulant graph is called cyclically almost self-complementary if it has an isomorphic almost complement with the same regular cyclic subgroup of the automorphism group. In this paper we prove that a cyclically almost self-complementary circulant of order 2n exists if and only if every prime divisor of n is congruent to 1 modulo 4, thus extending the known result on the existence of self-complementary circulants. We also describe the structure of cyclically almost self-complementary circulants and the action of their automorphism groups. Finally, we exhibit a class of almost self-complementary Cayley graphs on a dihedral group that are isomorphic to cyclically almost self-complementary circulants.

A few properties of circulant graphs: Self-complementary, isomorphism, Cartesian product and factorization

2017 7th International Conference on Modeling, Simulation, and Applied Optimization (ICMSAO), 2017

We present different properties of circulant graphs that includes (i) On the existence of self-complementary circulant graphs; (ii) Type-2 isomorphism, a new type of isomorphism other than already known Adam's isomorphism of circulant graphs and new abelian groups; (iii) Cartesian product and factorization of circulant graphs similar to the theory of product and factorization of natural numbers.

Circulant Graphs without Cayley Isomorphism Property with m j = 7

2016

A circulant graph Cn(R) is said to have the Cayley Isomorphism (CI) property if whenever Cn(S) is isomorphic to Cn(R), there is some a∈Zn ∗ for which S = aR. In this paper, we prove that for 1  n, 3  k, 1  i  7, di= 7n(i-1)+1 and Ri= {di , 49n-di , 49n+di , 98n-di , 98n+di , . . . , 294n-di , 294n+di , 343n-di , 7p1, 7p2, . . . , 7pk-2}, graphs C343n(Ri) are circulant without CI-property with mj = gcd(343n, rj ) = 7, rjRi , gcd(p1,p2,...,pk-2) = 1 and n,p1,p2,...,pk-2N. -----------------------------------------------------------AMS Subject Classification: 05C60, 05C25.

Circulant Graphs without Cayley Isomorphism Property with = 7

2016

A circulant graphCn (R) is said to have the Cayley Isomorphism (CI) property if whenever Cn (S)is isomorphic toCn (R), there is some a∈Zn ∗ for which S = aR.In this paper, we prove that C27n (R), C27n (S)and C27n (T)are isomorphic circulant graphs without CI-property whereR = {1, 9n-1, 9n+1, 3p1, 3p2, . . . , 3pk−2}, S = {3n+1, 6n-1, 12n+1, 3p1, 3p2, . . . , 3pk−2},T = {3n-1, 6n+1,12n-1, 3p1, 3p2, . . . ,3pk−2},k ≥ 3, gcd(p1, p2,..., pk−2) = 1 and n, p1,p2,...,pk−2Nand also obtain new abelian groups from these isomorphic circulant graphs. -----------------------------------------------------------AMS Subject Classification: 05C60, 05C25.

Circulant Graphs without Cayley Isomorphism Property with𝒎 = 3

IOSR Journals , 2019

A circulant graph𝐶𝑛 (𝑅) is said to have the Cayley Isomorphism (CI) property if whenever 𝐶𝑛 (𝑆)is isomorphic to𝐶𝑛 (𝑅), there is some a∈𝑍𝑛 ∗ for which S = aR.In this paper, we prove that 𝐶27𝑛 (𝑅), 𝐶27𝑛 (𝑆)and 𝐶27𝑛 (𝑇)are isomorphic circulant graphs without CI-property whereR = {1, 9n-1, 9n+1, 3𝑝1 , 3𝑝2 , . . . , 3𝑝𝑘−2 }, S = {3n+1, 6n-1, 12n+1, 3𝑝1 , 3𝑝2 , . . . , 3𝑝𝑘−2 },T = {3n-1, 6n+1,12n-1, 3𝑝1 , 3𝑝2 , . . . ,3𝑝𝑘−2 },k ≥ 3, gcd(𝑝1 , 𝑝2 ,..., 𝑝𝑘−2 ) = 1 and 𝑛, 𝑝1 ,𝑝2 ,...,𝑝𝑘−2ℕand also obtain new abelian groups from these isomorphic circulant graphs.

On Non-Cayley Tetravalent Metacirculant Graphs

Graphs and Combinatorics, 2002

In connection with the classification problem for non-Cayley tetravalent metacirculant graphs, three families of special tetravalent metacirculant graphs, denoted by U 1 , U 2 and U 3 , have been defined . It has also been shown [11] that any non-Cayley tetravalent metacirculant graph is isomorphic to a union of disjoint copies of a non-Cayley graph in one of the families U 1 , U 2 or U 3 . A natural question raised from the result is whether all graphs in these families are non-Cayley. We have proved recently in [12] that every graph in U 2 is non-Cayley. In this paper, we show that every graph in U 1 is also a connected non-Cayley graph and find an infinite class of connected non-Cayley graphs in the family U 3 .

On the Automorphism Group of Integral Circulant Graphs

The Electronic Journal of Combinatorics, 2011

The integral circulant graph X n (D) has the vertex set Z n = {0, 1, 2,. .. , n − 1} and vertices a and b are adjacent, if and only if gcd(a − b, n) ∈ D, where D = {d 1 , d 2 ,. .. , d k } is a set of divisors of n. These graphs play an important role in modeling quantum spin networks supporting the perfect state transfer and also have applications in chemical graph theory. In this paper, we deal with the automorphism group of integral circulant graphs and investigate a problem proposed in [W. Klotz, T. Sander, Some properties of unitary Cayley graphs, Electr. J. Comb. 14 (2007), #R45]. We determine the size and the structure of the automorphism group of the unitary Cayley graph X n (1) and the disconnected graph X n (d). In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups Aut(X n (1, p)) for n being a square-free number and p a prime dividing n, and Aut(X n (1, p k)) for n being a prime power.

Soumya 2018 On the automorphism groups of almost all circulant graphs and digraphs

We attempt to determine the structure of the automorphism group of a generic circulant graph. We first show that almost all circulant graphs have automorphism groups as small as possible. The second author has conjectured that almost all of the remaining circulant (di)graphs (those whose automorphism group is not as small as possible) are normal circulant (di)graphs. We show this conjecture is not true in general, but is true if we consider only those circulant (di)graphs whose order is in a “large” subset of integers. We note that all non-normal circulant (di)graphs can be classified into two natural classes (generalized wreath products, and deleted wreath type), and show that neither of these classes contains almost every non-normal circulant digraph.

On cycles in the sequence of unitary Cayley graphs

Discrete mathematics, 2004

For n ∈ N let p k (n) be the number of induced k-cycles in the Cayley graph Cay (Zn; Un), where Zn is the ring of integers mod n and Un = Z * n is the group of units mod n. Our main result is: Given r ∈ N there is a number m(r), depending only on r, with r ln r 6 m(r) 6 9r! such that p k (n) = 0 if k ¿ m(r) and n has at most r prime divisors. As a corollary we deduce the existence of non-trivial arithmetic functions f with the properties: f is a Z-linear combination of multiplicative arithmetic functions. f(n) = 0 for every n with at most r di erent prime divisors. We also prove the chromatic uniqueness of Cay (Zn; Un) for n a prime power.