We show that computing (and even approximating) MAXIMUM CLIQUE and MINIMUM GRAPH COLORING for circulant graphs is essentially as hard as in the general case. In contrast, we show that, under additional constraints, e.g., prime order and/or sparseness, GRAPH ISOMORPHISM and MINIMUM GRAPH COLORING become easier in the circulant case, and we take advantage of spectral techniques for their efficien...

We describe an approach to the circulant Hadamard conjecture based on Walsh-Fourier analysis. We show that the existence of a circulant Hadamard matrix of order n is equivalent to the existence of a non-trivial solution of a certain homogenous linear system of equations. Based on this system, a possible way of proving the conjecture is proposed.

We present the results concerning the embedding of trees into recursive circulants. Recursive circulant G(N,:,/) is a circulant graph with N vertices and jumps of powers of d. We present dilation 1 embedding of Fibonacci trees and full quaternaly trees in G(Z”, 2), and full binary trees and hinomial trees in G(2”,4).

Various results on factorisations of complete graphs into circulant graphs and on 2factorisations 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.

A classification of all extremal double circulant self-dual codes of lengths up to 72 is known. In this paper, we give a classification of all extremal double circulant self-dual codes of lengths 74–88. © 2006 Elsevier B.V. All rights reserved.

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.

A set of fast real transforms including the well known Hartley transform is fully investigated. Mixed radix splitting properties of Hartley-type transforms are examined in detail. The matrix algebras diagonalized by the Hartley-type matrices are expressed in terms of circulant and (−1)-circulant matrices. © 2002 Elsevier Science Inc. All rights reserved.