Let \(M\) be a finite dimensional commutative nilpotent algebra over a field \(K\) of prime characteristic \(p\). It has been conjectured that \(\dim M \geq p \;\dim M^{(p)}\), where \(M^{(p)}\) is the subalgebra of \(M\) generated by \(x^p\), \(x \in M\), \([2]\).This was proved (by Eggert) in the case \(\dim M^{(p)} \leq 2\) in \(1971\). This result was extended to the noncommutative case in \(1994\) \([8]\). Not only is this conjecture important in its own right, but it was shown (by Eggert) that a proof of the above conjecture would result in a complete classification of the group of units of finite commutative rings of characteristic \(p\) with an identity. In this short paper, we obtain a proof of Eggert’s conjecture in the case \(\dim M^{(p)} = 3\).

Černý, Horák, and Wallis introduced a generalization of Kirkman’s Schoolgirl Problem to the case where the number of schoolgirls is not a multiple of three; they require all blocks to be of size three, except that each resolution class should contain either one block of size two (when \(v \equiv 2 \pmod{3}\)) or one block of size four (when \(v \equiv 1 \pmod{3}\)). We consider the problem of determining the maximum (resp. minimum) possible number of resolution classes such that any pair of elements (schoolgirls) is covered at most (resp. at least) once.

A simple undirected graph \(G\) is called a \emph{sum graph} if there exists a labelling \(\lambda\) of the vertices of \(G\) into distinct positive integers such that any two distinct vertices \(u\) and \(v\) of \(G\) are adjacent if and only if there is a vertex \(w\) whose label \(\lambda(w) = \lambda(u) + \lambda(v)\). It is obvious that every sum graph has at least one isolated vertex, namely the vertex with the largest label. The \emph{sum number} \(\sigma(H)\) of a connected graph \(H\) is the least number \(r\) of isolated vertices \(\overline{K}_r\) such that \(G = H + \overline{K}_r\) is a sum graph.
It is clear that if \(H\) is of size \(m\), then \(\sigma(H) \leq m\). Recently, Hartsfield and Smyth showed that for wheels \(W_n\) of order \(n+1\) and size \(m = 2n\), \(\sigma(W_n) \in \Theta(m)\); that is, that the sum number is of the same order of magnitude as the size of the graph. In this paper, we refine these results to show that for even \(n \geq 4\), \(\sigma(W_n) = {n}/{2} + 2\), while for odd \(n \geq 5\) we disprove a conjecture of Hartsfield and Smyth by showing that \(\sigma(W_n) = n\). Labellings are given that achieve these minima.

We give a new algorithm which allows us to construct new sets of sequences with entries from the commuting variables \(0, \pm a, \pm b\), with zero autocorrelation function.We show that for eight cases if the designs exist they cannot be constructed using four circulant matrices in the Goethals-Seidel array. Furthermore, we show that the necessary conditions for the existence of an \(\text{OD}(44; s_1, s_2)\) are sufficient
except possibly for the following \(8\) cases:
\begin{align*}
(5,34), (8,31), (9,33), (13,29),\\
(7,32), (9,30), (11,30), (15,26)
\end{align*}
which could not be found because of the large size of the search space for a complete search. These cases remain open. In all we find \(399\) cases, show \(67\) do not exist
and establish \(8\) cases cannot be constructed using four circulant matrices.
We give a new construction for \(\text{OD}(2n)\) and \(\text{OD}(n+1)\) from \(\text{OD}(n)\).

We note that all \(\text{OD}(44; s_1, 44-s_2)\) are known except for \(\text{OD}(44; 16, 28)\). These give \(21\) equivalence classes of Hadamard matrices.

In this paper, we review combinatorial models for secret sharing schemes. A detailed comparison of several existing combinatorial models for secret sharing schemes is conducted. We pay particular attention to the ideal instances of these combinatorial models. We show that the models under examination have a natural hierarchy, but that the ideal instances of these models have a different hierarchy. We demonstrate that, in the ideal case, the combinatorial structures underlying the combinatorial models are essentially independent of the model being used. Furthermore, we show that the matroid
associated with an ideal scheme is uniquely determined by the access structure of the scheme and is independent of the model being used. Using this result, we present a combinatorial classification of
ideal threshold schemes.

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.