SUMMARY Strang's proposal to use a circulant preconditioner for linear systems of equations with a Hermitian positive definite Toeplitz matrix has given rise to considerable research on circulant preconditioners. This paper presents an {e iϕ }-circulant Strang-type preconditioner.

A circulant matrix of order n is the matrix of convolution by a fixed element of the group algebra of the cyclic group Zn. Replacing Zn by an arbitrary finite group G gives the class of matrices that we call G-circulant. We determine the eigenvalues of such matrices with the tools of representation theory and the non-abelian Fourier transform. Definition 1 An n by n matrix C is circulant if the...

In signal acquisition, Toeplitz and circulant matrices are widely used as sensing operators. They correspond to discrete convolutions and are easily or even naturally realized in various applications. For compressive sensing, recent work has used random Toeplitz and circulant sensing matrices and proved their efficiency in theory, by computer simulations, as well as through physical optical exp...

We consider monotonicity problems for graph searching games. Variants of these games – defined by the type of moves allowed for the players – have been found to be closely connected to graph decompositions and associated width measures such as pathor tree-width. Of particular interest is the question whether these games are monotone, i.e. whether the cops can catch a robber without ever allowin...

The necessary condition for eigenvalue values of a circulant matrix is studied It is then proved that the necessary condition also su ces the existence of a circulant matrix with the prescribed eigenvalue values Introduction An n n matrix C of the form C c c cn cn c c cn cn cn c cn c c cn c is called a circulant matrix As each row of a circulant matrix is just the previous row cycled forward on...

The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini...

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skewcirculant and g-circulant matrices. The entries are product of binomial coefficients with Fibonacci numbers and Lucas numbers, respectively. We obtain identity estimations for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

We explore ways to concisely describe circulant graphs, highly symmetric graphs with properties that are easier to generalize than other graphs. We introduce connection sets as a way to describe isomorphism among some circulant graphs and discuss Ádám’s conjecture that attempts to generalize this idea to all circulant graphs, exploring cases in which the conjecture holds and cases in which it f...

In this paper, the total chromatic number and fractional total chromatic number of circulant graphs are studied. For cubic circulant graphs we give upper bounds on the fractional total chromatic number and for 4-regular circulant graphs we find the total chromatic number for some cases and we give the exact value of the fractional total chromatic number in most cases.