On the ' Ad'am Property for Circulant Graphs (original) (raw)

Graphs with circulant adjacency matrices

Journal of Combinatorial Theory, 1970

Properties of a graph (directed or undirected) whose adjacency matrix is a circulant are studied. Examples are given showing that the connection set determined by the first row of such a matrix need not be multiplicatively related to the connection set of an isomorphic graph. Two different criteria are given under which two graphs with circulant adjacency matrices are isomorphic if and only if their connection sets are multiplicatively related. The first criterion is that the graphs have a prime number of vertices. The second criterion is that the adjacency matrices have non-repeated eigenvalues. The final section gives a partial characterization of graphs with n vertices whose automorphism group is the cyclic group C~.

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.

On the spectral Ádám property for circulant graphs

Discrete Mathematics, 2002

We investigate a certain condition for isomorphism between circulant graphs (known as the à Adà am property) in a stronger form by referring to isospectrality rather than to isomorphism of graphs. We describe a wide class of graphs for which the à Adà am conjecture holds. We apply these results to establish an asymptotic formula for the number of non-isomorphic circulant graphs and connected circulant graphs.

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.

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 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.

On Routing in Circulant Graphs of Degree Four

2005

In this paper we present the first polynomial time deterministic algorithm to compute the shortest path between two vertices of a circulant graph of degree four. Our spectacular algorithm only requires O(log 3 N ) bit operations, where N is the number of the vertices and it is based on shortest vector problems in a special class of lattices for L 1 -norm. Moreover, the technique can be extended to weighted and directed circulant graphs, the so called double-loop networks. Our main tools are results and methods from the geometry of numbers and computer algebra.

On Covering the Nodes of Circulant Networks and Its Applications

Coverage and security of network nodes are the critical issue in wireless sensor networks. The covering number is an essential tool in monitoring networks, as it gives the smallest number of detection devices needed to identify the location of an intruder or a malicious node. In this paper, we provide a graph theory based approach which could be beneficial to keep the network secure. The covering number for circulant networks are calculated exactly and their theoretical properties are explored in this article. Circulant networks have a broad range of applications in parallel computing and signal processing because of its increased connectivity. The minimal vertex covering set of a network serves as its back bone in the process of routing.

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.