Isomorphism of circulant graphs and digraphs (original) (raw)

On the Ádám Conjecture on Circulant Graphs

Lecture Notes in Computer Science, 1998

We investigate the condition for isomorphism between circulant graphs which is known as theÁdám property. We describe a wide class of graphs for which theÁdám conjecture holds (and even in a stronger form refering to isospectrality rather than to isomorphism of graphs.

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 the automorphism groups of almost all circulant graphs and digraphs

Ars Mathematica Contemporanea

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.

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.

VERTEX-TRANSITIVE GRAPHS OF ORDER 2p

Annals of the New York Academy of Sciences, 1979

In the past ten years there has been a considerable amount of activity in the area of circulant graphs and digraphs. Most of this has consisted of investigation of basic properties of circulants along with some applications. We shall now summarize some of this activity.

A note on arc-transitive circulant digraphs

2009

Abstract We prove that, for a positive integer n and subgroup H of automorphisms of a cyclic group Z of order n, there is up to isomorphism a unique connected circulant digraph based on Z admitting an arc-transitive action of Z⋊ H. We refine the Kovács–Li classification of arc-transitive circulants to determine those digraphs with automorphism group larger than Z⋊ H.

Distinguishing Number for some Circulant Graphs

Introduced by Albertson et al. \cite{albertson}, the distinguishing number D(G)D(G)D(G) of a graph GGG is the least integer rrr such that there is a rrr-labeling of the vertices of GGG that is not preserved by any nontrivial automorphism of GGG. Most of graphs studied in literature have 2 as a distinguishing number value except complete, multipartite graphs or cartesian product of complete graphs depending on nnn. In this paper, we study circulant graphs of order nnn where the adjacency is defined using a symmetric subset AAA of mathbbZn\mathbb{Z}_nmathbbZn, called generator. We give a construction of a family of circulant graphs of order nnn and we show that this class has distinct distinguishing numbers and these lasters are not depending on nnn.

On isomorphic factorizations of circulant graphs

Journal of Combinatorial …, 2006

Brian Alspach,1 Danny Dyer,1 Donald L. Kreher2 1Department of Mathematics and Statistics, University of Regina, Regina, Saskatchewan, Canada S42 0A2, E-mail: alspach@math.uregina.ca ... 2Department of Mathematical Sciences, Michigan ...