We describe several techniques for constructing \(n\)-dimensional Hadamard matrices from \(2\)-dimensional Hadamard matrices, and note that they may be applied to any perfect binary array \((PBA)\), thus optimally improving a result of Yang. We introduce cocyclic perfect binary arrays, whose energy is not restricted to being a perfect square. These include all of Jedwab’s generalized perfect binary arrays. There are many more cocyclic \(PBAs\) than \(PBAs\). We resolve a potential ambiguity inherent in the “weak difference set” construction of \(n\)-dimensional Hadamard matrices from cocyclic \(PBAs\) and show it
is a relative difference set construction.

Two graphs are matching equivalent if they have the same matching polynomial. We prove that several infinite families of pairs of graphs are pairwise matching equivalent. We also establish some divisibility relations among matching polynomials. Furthermore, we demonstrate that the matching polynomials of certain graphs serve as a polynomial model for the Fibonacci numbers and the Lucas numbers.

In this paper, we establish necessary and sufficient conditions on \(m\) and \(n\) in order for \(K_m \times K_n\), the Cartesian product of two complete graphs, to be decomposable into cycles of length \(4\). The main result is that \(K_m \times K_n\) can be decomposed into cycles of length \(4\) if and only if either \(m, n \equiv 0 \pmod{2}\), \(m, n \equiv 1 \pmod{8}\), or \(m, n \equiv 5 \pmod{8}\).

This paper contributes to the determination of all integers of the form \(pqr\), where \(p\), \(q\), and \(r\) are distinct odd primes, for which there exists a vertex-transitive graph on \(pqr\) vertices that is not a Cayley graph. The paper addresses the situation where there exists a vertex-transitive subgroup \(G\) of automorphisms of such a graph which has a chain \(1 < N < K < G\) of normal subgroups, such that both \(N\) and \(K\) are intransitive on vertices and the \(N\)-orbits are proper subsets of the \(K\)-orbits.

We discuss difference sets (DS) and supplementary difference sets (SDS) over rings. We survey some constructions of SDS over Galois rings where there are no short orbits. From there, we move to constructions involving short orbits, yielding new infinite families of SDS over \(\text{GF}(p) \times \text{GF}(q)\), \(p\), \(q\) both prime powers.Many of these families have \(\lambda = 1\). We also present new balanced incomplete block designs and pairwise balanced designs arising from the constructions given here.

Using a blend of Drake’s and Saha’s techniques, we construct a \(\text{BTD}(n^2/4; (n^2 + n)/2; 2n – 4, 3, 2n + 2; n; 8)\) whenever \(n\) is a power of \(2\), as well as some new symmetric \(\text{BTDs}\).It is known that the necessary condition \(v \equiv 1 \pmod{2}\) is sufficient for the existence of simple \(\text{BIBD}(v, 3, 3)\).In the second part of this paper, we provide a simple construction based on graph factorization to prove this result whenever \(v\) is not divisible by \(3\).We then expand upon this result to exhibit further constructions of \(\text{BTDs}\).

We consider the projective properties of small Hadamard matrices when viewed as two-level \(OAs\) of strength two. We show that in some cases sets of rows with the same type of projection form balanced incomplete block designs.

Let \(H_i\) be the \(3\)-uniform hypergraph on \(4\) vertices with \(i\) hyperedges. In this paper, we settle the existence of \(H_3\)-hypergraph designs of index \(\lambda\), obtaining simple \(H_3\)-hypergraph designs when \(\lambda = 2\), and providing a new proof of their existence when \(\lambda = 1\). The existence of simple \(H_2\)-hypergraph designs of index \(\lambda\) is completely settled, as is the spectrum of \(H_2\)-hypergraph designs of index \(\lambda\).

This paper provides a general method for finding a critical set for any Latin square of order \(n\). This method is used to prove the existence of critical sets of various sizes. It has also been applied to all main classes of Latin squares of order seven, thereby producing a critical set for each Latin square of order seven.