The aim of this work is twofold: to unify, systematize and extend the known results of the analytical (i.e. formula-wise) counting of directed and undirected circulant graphs with a prime or, more generally, square-free number of vertices; to develop a general combinatorial framework for the counting of nonisomorphic circulant graphs with pk vertices, p odd prime, k 2. The general problem of co...

The intention of the paper is to move a step towards a classification of network topologies that exhibit periodic quantum dynamics. We show that the evolution of a quantum system whose hamiltonian is identical to the adjacency matrix of a circulant graph is periodic if and only if all eigenvalues of the graph are integers (that is, the graph is integral). Motivated by this observation, we focus...

The standard products—cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by W. Imrich and H. Izbicki (1975, Monatsh. Math. 80, 277–281) and later called B-products by I. Broere and J. H. Hattingh (1990, Quaest. Math. 13, 191–216) who establish that the lexicographic product of two circulant graphs is again circulant. In this paper, we establish that any B-...

In this paper circulant matrices in max-plus algebra are presented. Circulant matrices are special form of matrices which are entered by vector of inputs. For special types of matrices such as circulant matrices, the computation can often be performed in the simpler way than in the general case. The so-called max-plus algebra is useful for investigation of discrete events systems and the sequen...

Given a matrix M of size n, the digraph D on n vertices is said to be the digraph of M , when Mij 6= 0 if and only if (vi, vj) is an arc of D. We give a necessary condition, called strong quadrangularity, for a digraph to be the digraph of a unitary matrix. With the use of such a condition, we show that a line digraph, − → L D, is the digraph of a unitary matrix if and only if D is Eulerian. It...

Let D be a digraph, V (D) and A(D) will denote the sets of vertices and arcs of D, respectively. A (k, l)-kernel N of D is a k-independent (if u, v ∈ N then d(u, v), d(v, u) ≥ k) and l-absorbent (if u ∈ V (D) − N then there exists v ∈ N such that d(u, v) ≤ l) set of vertices. A k-kernel is a (k, k − 1)-kernel. A digraph D is transitive if (u, v), (v, w) ∈ A(D) implies that (u,w) ∈ A(D). This co...

The main result is the “black dot algorithm” and its fast version for the construction of a new circulant preconditioner for Toeplitz matrices. This new preconditioner C is sought directly as a solution to one of possible settings of the approximation problem A ≈ C + R, where A is a given matrix and R should be a “low-rank” matrix. This very problem is a key to the analysis of superlinear conve...

In this paper, a recently introduced block circulant preconditioner for the linear systems of the codes for ordinary differential equations (ODEs) is investigated. Most ODE codes based on implicit formulas, at each integration step, need the solution of one or more unsymmetric linear systems that are often large and sparse. Here, the boundary value methods, a class of implicit methods for the n...

We study the relationship between two key concepts in the theory of digraphs, those of quotient digraphs and voltage digraphs. These techniques contract or expand a given digraph in order to study its characteristics, or obtain more involved structures. As an application, we relate the spectrum of a digraph Γ, called voltage digraph or base, with the spectrum of its lifted digraph Γ. We prove t...