On isomorphic factorizations of circulant graphs (original) (raw)
A Study on Isomorphic Properties of Circulant Graphs
Advances in Science, Technology and Engineering Systems Journal, 2017
C n (R) denotes circulant graph C n (r 1 , r 2 , . . . , r k ) of order n for a set R = {r 1 , r 2 , . . . , r k } where n for which S = aR. In this paper, isomorphic properties of circulant graphs that includes (i) Self-complementary circulant graphs; (ii) Type-2 isomorphism, a new type of isomorphism other than already known Adam's isomorphism of circulant graphs and (iii) Cartesian product and factorization of circulant graphs similar to the theory of product and factorization of natural numbers are studied. New abelian groups are obtained from these isomorphic circulant graphs. Type-2 isomorphic circulant graphs have the property that they are isomorphic graphs without Cayley Isomorphism (CI) property and thereby new families.
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.
On factorisations of complete graphs into circulant graphs and the Oberwolfach Problem
Various results on factorisations of complete graphs into circulant graphs and on 2-factorisations of these circulant graphs are proved. As a consequence, a number of new results on the Oberwolfach Problem are obtained. For example, a complete solution to the Oberwolfach Problem is given for every 2-regular graph of order 2p where p = 5 (mod 8) is prime.
Isomorphism of circulant graphs and digraphs
Discrete Mathematics, 1979
Let Sc{l,...,n-1) satisfy -S =S mod n. The circulant graph G(n, S) with vertex set IQ, IJ l,"', u,_r} and edge set E satisfies "iiIj E E if and only if i -i E S, where all arithmetic is done mod n. The circulant digraph G(n, S) is defined similarly without the restriction S = -S. Adarn conjectured that G(n, S)=G(n, S') if and only if S = US' for some unit u mod n. In this paper we prove the conjecture true if n = pq where p and q are distinct primes. We also show that it is not generally true when n = p2, and determine exact conditions on S that it be tme in this case. We then show as a simple consequence that the conjecture is false in most cases when n is divisible by p2 where p is an odd prime, or n is divisible by 24.
On New Type of Isomorphic Circulant Graphs of Order np3np^3np3 and New Groups
arXiv: Combinatorics, 2020
Circulant graphs C n (R) and C n (S) are said to be Adam's isomorphic if there exist some a ∈ Z * n such that S = aR. A circulant graph C n (R) is said to have the Cayley Isomorphism (CI) property if whenever C n (S) is isomorphic to C n (R), there is some a ∈ Z * n for which S = aR. Vilfred [19] defined and studied Type-2 circulant graph isomorphism, a new type of isomorphism different from Adam's isomorphism and without CI-property and we obtained families of isomorphic circulant graphs of Type-2 with respect to r = 2,3,5,7 [19], [21]-[23]. In this paper, we obtain new families of Type-2 isomorphic circulant graphs with respect to r = p and of order np 3 and new abelian groups on these isomorphic graphs where p is a prime number and n ∈ N. Theorems 4.2 and 4.4 are the main results. Using Theorem 4.4, a list of new abelian groups, (T 2 np 3 ,p (C np 3 (R np 3 ,x+yp i)), •) are given in the annexure for p = 3,5,7,11 and n = 1 to 5 and also for p = 13 and n = 1 to 3, 1 ≤ x ≤ p − 1 and 0 ≤ y ≤ np − 1, p, np 3 − p ∈ R np 3 ,x+yp i where T 2 np 3 ,p (C np 3 (R np 3 ,x+yp i)) = {C np 3 (R np 3 ,x+yp i) : i = 1 to p} is a set of Type-2 isomorphic circulant graphs C np 3 (R np 3 ,x+yp i
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.
Some Invariants of Circulant Graphs
Symmetry, 2016
Topological indices and polynomials are predicting properties like boiling points, fracture toughness, heat of formation, etc., of different materials, and thus save us from extra experimental burden. In this article we compute many topological indices for the family of circulant graphs. At first, we give a general closed form of M-polynomial of this family and recover many degree-based topological indices out of it. We also compute Zagreb indices and Zagreb polynomials of this family. Our results extend many existing results.
Journal of Combinatorial Theory, Series B, 2019
The canonical double cover D(Γ) of a graph Γ is the direct product of Γ and K 2. If Aut(D(Γ)) = Aut(Γ) × Z 2 then Γ is called stable; otherwise Γ is called unstable. An unstable graph is nontrivially unstable if it is connected, non-bipartite and distinct vertices have different neighborhoods. In this paper we prove that every circulant graph of odd prime order is stable and there is no arctransitive nontrivially unstable circulant graph. The latter answers a question of Wilson in 2008. We also give infinitely many counterexamples to a conjecture of Marušič, Scapellato and Zagaglia Salvi in 1989 by constructing a family of stable circulant graphs with compatible adjacency matrices.