A general algorithm for finding a minimal critical set for any latin square is presented. By implementing this algorithm, minimal critical sets for all the latin squares of order six have been found. In addition, this algorithm is used to prove that the size of the minimal critical set for a back circulant latin square of order seven is twelve, and for order nine is twenty. These results provid...

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms o...

A set S of vertices of a graph G is a dominating set of G if every vertex u of G is either in S or it has a neighbour in S. In other words S is dominating if the sets S ∩N [u] where u ∈ V (G) and N [u] denotes the closed neighbourhood of u in G, are all nonempty. A set S ⊆ V (G) is called a locating code in G, if the sets S ∩ N [u] where u ∈ V (G) \ S are all nonempty and distinct. A set S ⊆ V ...

Let G = (V,E) be a graph of order n and size e. An (a, d)-vertexantimagic total labeling is a bijection α from V (G) ∪ E(G) onto the set of consecutive integers {1, 2, . . . , n+ e}, such that the vertex-weights form an arithmetic progression with the initial term a and the common difference d. The vertex-weight of a vertex x is the sum of values α(xy) assigned to all edges xy incident to the v...

We employ theoretical and computational techniques to construct new weighing matrices constructed from two circulants. In particular, we construct W (148, 144), W (152, 144), W (156, 144) which are listed as open in the second edition of the Handbook of Combinatorial Designs. We also fill a missing entry in Strassler’s table with answer ”YES”, by constructing a circulant weighing matrix of orde...

The feedback delay network (FDN) has been proposed for digital reverberation. Also proposed with similar advantages is the digital waveguide network (DWN). This paper notes that the commonly used FDN with an N × N orthogonal feedback matrix is isomorphic to a normalized digital waveguide network consisting of one scattering junction and a vector transformer joining N reflectively terminated bra...

Reliability and efficiency are important criteria in the design of interconnection networks. Recently, the w-wide diameter dw(G), the (w − 1)-fault diameter Dw(G), and the w-Rabin number rw(G) have been used to measure network reliability and efficiency. In this paper, we study dw(G), Dw(G) and rw(G) using the strong w-Rabin number r∗ w(G) for 1 ≤ w ≤ k(G) and G is a circulant network G(dn; {1,...

In this paper we consider the problem of describing the possible exponents of boolean primitive circulant matrices. We give a conjecture for the possible such exponents and prove this conjecture in several cases. In particular, we consider in greater detail the case of matrices whose generating vector has three nonzero entries.

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...

Article history: Received 28 December 2012 Accepted 12 June 2013 Available online xxxx Submitted by A. Böttcher MSC: 15A83 15B05 93E12 60G10 49N15 65K10