A number of new weighing matrices constructed from two circulants and via a direct sum construction are presented, thus resolving several open cases for weighing matrices as these are listed in the second edition of the Handbook of Combinatorial Designs.

A 1-factorisation of a graph G is a decomposition of G into edge-disjoint 1-factors (perfect matchings), and a perfect 1-factorisation is a 1-factorisation in which the union of any two of the 1-factors is a Hamilton cycle. We consider the problem of the existence of perfect 1-factorisations of even order circulant graphs with small degree. In particular, we characterise the 3-regular circulant...

In this paper, it is considered a g-circulant, right circulant, left circulant and a special kind of tridiagonal matrices whose entries are h(x)-Fibonacci polynomials. The determinant of these matrices is established and with the tridiagonal matrices we show that the determinant is equal to the nth term of the h(x)-Fibonacci polynomials.

There is a contrast between the two sets of functional equations f0(x + y) = f0(x)f0(y) + f1(x)f1(y), f1(x + y) = f1(x)f0(y) + f0(x)f1(y), and f0(x − y) = f0(x)f0(y)− f1(x)f1(y), f1(x − y) = f1(x)f0(y)− f0(x)f1(y) satisfied by the even and odd components of a solution of f(x + y) = f(x)f(y). J. Schwaiger and, later, W. Förg-Rob and J. Schwaiger considered the extension of these ideas to the cas...

Using the doubling lemma, amicable sets of eight circulant matrices and four complementary negacyclic matrices, we show that all full triples are types of orthogonal designs of order 56. This implies that all full orthogonal designs OD(2t7;x, y, 2t7− x− y) exist for any t ≥ 3.

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-...

A k-extended Skolem-type 5-tuple difference set of order t is a set of t 5-tuples {(di,1, di,2, di,3, di,4, di,5) | i = 1, 2, . . . , t} such that di,1+di,2+di,3+di,4+di,5 = 0 for 1 ≤ i ≤ t and {|di,j| | 1 ≤ i ≤ t, 1 ≤ j ≤ 5} = {1, 2, . . . , 5t+1}\{k}. In this talk, we will give necessary and sufficient conditions on t and k for the existence of a k-extended Skolem-type 5-tuple difference set ...

We define the notions of nest and influence of a subset of a critical set of a back circulant latin square, and study their properties. We also show that a secret sharing scheme based on a critical set of a latin square is both compartmentalised, and hierarchical.

Note. The determinant of a circulant matrix is an example of a group determinant, where the group is the cyclic group of order n. In 1880 Dedekind suggested generalizing the case of circulants (and more generally group de­ terminants for abelian groups) to arbitrary groups. It was this suggestion that led Frobenius to the creation group of representation theory. See [1] and the references therein.