On the Ádám Conjecture on Circulant Graphs (original) (raw)
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 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 ...
arXiv (Cornell University), 2017
The Zykov ring of signed finite simple graphs with topological join as addition and compatible multiplication is an integral domain but not a unique factorization domain. We know that because by the graph complement operation it is isomorphic to the strong Sabidussi ring with disjoint union as addition. We prove that the Euler characteristic is a ring homomorphism from the strong ring to the integers by demonstrating that the strong ring is homotopic to a Stanley-Reisner Cartesian ring. More generally, the Kuenneth formula holds on the strong ring so that the Poincaré polynomial is compatible with the ring structure. The Zykov ring has the clique number as a ring homomorphism. Furthermore, the Cartesian ring has the property that the functor which attaches to a graph the spectrum of its connection Laplacian is multiplicative. The reason is that the connection Laplacians do tensor under multiplication, similarly to what the adjacency matrix does for the weak ring. The strong ring product of two graphs contains both the weak and direct product graphs as subgraphs. The Zykov, Sabidussi or Stanley-Reisner rings are so manifestations of a network arithmetic which has remarkable cohomological properties, dimension and spectral compatibility but where arithmetic questions like the complexity of detecting primes or factoring are not yet studied well. We illustrate the Zykov arithmetic with examples, especially from the subring generated by point graphs which contains spheres, stars or complete bipartite graphs. While things are formulated in the language of graph theory, all constructions generalize to the larger category of finite abstract simplicial complexes.
On 4-valent Frobenius circulant graphs
Discrete Mathematics & Theoretical Computer Science
Graph Theory A 4-valent first-kind Frobenius circulant graph is a connected Cayley graph DLn(1, h) = Cay(Zn, H) on the additive group of integers modulo n, where each prime factor of n is congruent to 1 modulo 4 and H = {[1], [h], −[1], −[h]} with h a solution to the congruence equation x 2 + 1 ≡ 0 (mod n). In [A. Thomson and S. Zhou, Frobenius circulant graphs of valency four, J. Austral. Math. Soc. 85 (2008), 269-282] it was proved that such graphs admit 'perfect ' routing and gossiping schemes in some sense, making them attractive candidates for modelling interconnection networks. In the present paper we prove that DLn(1, h) has the smallest possible broadcasting time, namely its diameter plus two, and we explicitly give an optimal broadcasting in DLn(1, h). Using number theory we prove that it is possible to recursively construct larger 4-valent first-kind Frobenius circulants from smaller ones, and we give a methodology for such a construction. These and existing result...
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 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.
Resolvability in circulant graphs
Acta Mathematica Sinica, English Series, 2012
A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d (u, w) = d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d (u, s)
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.