In this paper, we investigate the reduced form of circulant matrices and we show that the problem of computing the q-th roots of a nonsingular circulant matrix A can be reduced to that of computing the q-th roots of two half size matrices B - C and B + C.

1. Introduction. A useful general strategy for the construction of interesting families of vertex-transitive graphs is to begin with some family of transitive permutation groups and to construct for each group T in the family all graphs G whose vertex-set is the orbit V of T and for which T ^ Aut (G), where Aut (G) denotes the automorphism group of G. For example, if we consider the family of c...

We take the first step towards establishing a theory of Parker vectors for infinite permutation groups, with an emphasis towards oligomorphic groups. We show that, on the one hand, many results for finite groups extend naturally to the infinite case (Parker’s Lemma, multiplicative properties, etc.), while on the other, in the infinite case some genuinely new phenomena arise. We also note that c...

Let G = (V, E) be a graph without isolated vertices. A set D ⊆ V is a d-distance paired-dominating set of G if D is a d-distance dominating set of G and the induced subgraph 〈D〉 has a perfect matching. The minimum cardinality of a d-distance paired-dominating set for graph G is the d-distance paired-domination number, denoted by γd p(G). In this paper, we study the ddistance paired-domination n...

In a given graph G, a set of vertices S with an assignment of colors is said to be a defining set o f the vertex coloring o f G, if there exists a unique extension of the colors of S to a z(G)coloring of the vertices of G. The concept of a defining set has been studied, to some extent, for block designs and also under another name, a critical set, for latin squares. In this note we extend this ...

In 1994, Cornuéjols and Novick published a classification of ideal and minimally non ideal circulant clutters. One of their main results for doing so relates contractions of these clutters, simple directed cycles in an appropriate graph, and algebraic conditions. The purpose of this paper is twofold: to correct a small inaccuracy of the necessity of the algebraic conditions in the original proo...

A graph X with at least two independent edges is 2-extendable if any two independent edges of X are contained in a perfect matching of X. In this paper, we prove that a connected Cayley graph of even order on a dihedral group is 2-extendable if and only if it is not isomorphic to anyone of the following circulant graphs: (I) Z2n(1, 2n 1), n ~ 3; (II) Z2n(1,2,2n-l,2n-2), n~3; (III) Z4n(1,4n -1,2...

It is known that if a bipartite graph G with n edges possesses any of three types of ordered labelings, then the complete graphK2nx+1 admits a cyclic G-decomposition for every positive integer x. We introduce variations of the ordered labelings and show that whenever a bipartite graph G admits one of these labelings, then there exists a cyclic G-decomposition of an infinite family of circulant ...

Let n 1 be an integer. The hypercube Qn is the graph whose vertex set isf0;1gn, where two n-tuples are adjacent if they differ in precisely one coordinate. This graph has many applications in Computer sciences and other area of sciences. Inthe graph Qn, the layer Lk is the set of vertices with exactly k 1’s, namely, vertices ofweight k, 1 k n. The hyper-star graph B(n;k) is...

We investigate the condition for isomorphism between circulant graphs which is known as the Ádám property. We describe a wide class of graphs for which the Ádám conjecture holds (and even in a stronger form refering to isospectrality rather than to isomorphism of graphs. In the important case of double loop (circulant graphs of degree 4) we give an almost complete classification of all possible...