Showing papers on "Circulant matrix published in 1986"

Connectivity of circulant digraphs

Abstract: An explicit expression is derived for the connectivity of circulant digraphs.

Super line-connectivity properties of circulant graphs

Abstract: The connection between line-connectivity concepts of graphs and indices of network reliability is well-known. Of particular interest in such studies are the circulant graphs because the connected ones have the largest possible value of line-connectivity $\lambda $ of p-point, degree r, regular graphs, namely $\lambda = r$. In this work, we define the higher order line-connectivity measure $N_i $ as the number of line-disconnecting sets of order i. Regular degree r, p-point graphs having $\lambda = r$ satisfy $N_\lambda \geqq p$. Such graphs which attain this lower bound are called super-$\lambda $. In this work we determine the necessary and sufficient conditions for a circulant to be super-$\lambda$. In addition we determine a lower bound on $N_i $ for $\lambda \leqq i\leqq 2r - 3$. It is shown that a special class of circulants, known as Harary graphs, achieve this lower bound for all these values of i.

The shapes of a random sequence of triangles

Abstract: A triangle with vertices z,, z2, z3 in the complex plane may be denoted by a vector Z, Z = [z, z2, z3]t. From a sequence of independent and identically distributed 3 x 3 circulants {(Ci}', we may generate from Z1 the sequence of vectors or triangles (Zj }', by the rule Zj = CjZji, (j> 1), Z, = Z. The 'shape' of a set of points, the simplest case being three points in the plane has been defined by Kendall (1984). We give several alternative, ab initio discussions of the shape of a triangle, and proofs of a limit theorem for shape of the triangles in the sequence {Zj} . In Appendix A, the shape concept is applied to the zeros of a cubic polynomial. Appendix B contains some further remarks about shape. Appendix C uses the methods of this paper to give proofs of generalizations of two old theorems on triangles. CIRCULANT; MOEBIUS TRANSFORMATIONS; NAPOLEON'S THEOREM

The relation between maximum likelihood estimation of structured covariance matrices and periodograms

Abstract: A generalized Burg technique has been developed recently by Burg, Luenberger, and Wegner for maximum likelihood estimation of structured covariance matrices. In this correspondence, the unique solution for the positive definite estimate over a class of nonnegative definite, symmetric matrices with known eigenvectors is presented. This solution coincides with the Karhunen-Loeve expansion, and for the class of circulant matrices can be interpreted in terms of periodograms. For stationary processes and infinitely large sample size, it is shown that the sequence of optimal covariance matrices among the class of circulant matrices is asymptotically equivalent to the sequence of true covariance matrices as the observation length approaches infinity.

Level- m scaled circulant factor matrices over the complex number field and the quaternion division algebra

Abstract: The level-m scaled circulant factor matrix over the complex number field is introduced. Its diagonalization and spectral decomposition and representation are discussed. An explicit formula for the entries of the inverse of a level-m scaled circulant factor matrix is presented. Finally, an algorithm for finding the inverse of such matrices over the quaternion division algebra is given.

On the Factorisation of Certain Symmetric Circulant Banded Linear Systems

Abstract: A factorisation method is described for the fast numerical solution of certain circulant banded symmetric linear systems which occur repeatedly in the numerical solution of differential equations. It can be shown that such special banded matrices Ar of semi-bandwidth r can be factorised into the product of easily inverted matrices, the components of which are a cyclic matrix and its transpose and a similar circulant banded matrix Ar-1 of order 1 less. By using this factorisation, efficient algorithmic solution methods can be derived for the related linear systems ([1],[2]).

Combinatorial Structures Corresponding to Reflective Circulant (0, 1)-Matrices

Abstract: Let C be a circulant (0, 1)-matrix and let us arrange the elements of the first row of C regularly on a circle. If there exists a diameter of the circle with respect to which 1's are symmetric, we call C reflective. In this paper we prove some properties of the reflective circulant (0, 1)-matrices and of certain corresponding combinatorial structures.