We associate to any given circulant complex matrix C another one E(C) such that E(E(C)) = C∗ the transpose conjugate of C. All circulant Hadamard matrices of order 4 satisfy a condition C4 on their eigenvalues, namely, the absolute value of the sum of all eigenvalues is bounded above by 4. We prove by a “descent” that uses our operator E that the only circulant Hadamard matrices of order n > 4,...

For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i− j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedba...

We investigate conditions for isomorphism between circulant graphs and analyze their automorphism group, with special emphasis on loop networks. We give a survey of pertinent published results and answer partially a long-standing conjecture [5] that all degree 4 circulants which are isomorphic are Ad am isomorphic. This paper also surveys solutions to several problems introduced in [9]. In part...

We study the maximum out forests of a (weighted) digraph and the matrix of maximum out forests. A maximum out forest of a digraph Γ is a spanning subgraph of Γ that consists of disjoint diverging trees and has the maximum possible number of arcs. If a digraph contains out arborescences, then maximum out forests coincide with them. We consider Markov chains related to a weighted digraph and prov...

A kernel N of a digraph D is an independent set of vertices of D such that for every w ∈ V (D)−N there exists an arc from w to N . If every induced subdigraph of D has a kernel, D is said to be a kernel perfect digraph. Minimal non-kernel perfect digraph are called critical kernel imperfect digraph. If F is a set of arcs of D, a semikernel modulo F , S of D is an independent set of vertices of ...

Abstract It is known that the spectral radius of a digraph with k edges is ≤ √ k, and that this inequality is strict except when k is a perfect square. For k = m + l, l fixed, m large, Friedland showed that the optimal digraph is obtained from the complete digraph on m vertices by adding one extra vertex and connecting it to the first l/2 vertices by pairs of directed edges. Using a combinatori...

We study spanning diverging forests of a digraph and related matrices. It is shown that the normalized matrix of out forests of a digraph coincides with the transition matrix in a specific observation model for Markov chains related to the digraph. Expressions are given for the Moore-Penrose generalized inverse and the group inverse of the Kirchhoff matrix. These expressions involve the matrix ...

The vertex set and arc set of a digraph D are denoted by V (D) and E (D), respectively, and the number of vertices in a digraph D is denoted by n (D). A directed cycle (path, walk) in a digraph will simply be called a cycle (path, walk). A graph or digraph is called hamiltonian if it contains a cycle that visits every vertex, traceable if it contains a path that visits every vertex, and walkabl...

In [18], Farrell and Whitehead investigate circulant graphs that are uniquely characterized by their matching and chromatic polynomials (i.e., graphs that are “matching unique” and “chromatic unique”). They develop a partial classification theorem, by finding all matching unique and chromatic unique circulants on n vertices, for each n ≤ 8. In this paper, we explore circulant graphs that are un...

We introduce circulant matrices that capture the structure of a skew-polynomial ring F[x; θ] modulo the left ideal generated by a polynomial of the type x − a. This allows us to develop an approach to skew-constacyclic codes based on such circulants. Properties of these circulants are derived, and in particular it is shown that the transpose of a certain circulant is a circulant again. This rec...