We give a new construction for skew-Hadamard matrices. This yields new infinite families of skew-Hadamard matrices, including 43 new skew-Hadamard matrices of order \(4q < 4000\).

The binary and ternary codes spanned by the rows of the point-by-block, pair-by-block, block-by-point incidence matrices of some 2-designs of small orders and their orthogonal complements are studied. Among some results, it is shown that if the code is properly chosen, then the weight distribution of the code serves as an appropriate design isomorphism invariant. The automorphism groups of the codes and the design are computed.

A Latin square of order \(n\) is an \(n \times n\) array of cells containing one of the \(n\) elements in \(\{1,2,\ldots,n\}\) such that in each row and each column each element appears exactly once. A partial transversal \(P\) of a Latin square \(L\) is a set of \(n\) cells such that no two are in the same row and the same column. The number of distinct elements in \(P\) is referred to as the length of \(P\), denoted by \(|P|\), and the maximum length of a partial transversal in \(L\) is denoted by \(t(L)\). In this paper, we study the technique used by Shor which shows that \(t(L) \geq n – 5.53{(\ln)}^2\) and we improve the lower bound slightly by using a more accurate evaluation.

The maximum possible toughness among graphs with \(n\) vertices and \(m\) edges is considered. This is an analog of the corresponding problem regarding maximum connectivity solved by Harary. We show that, if \(m < \lceil \frac{3n}{2} \rceil\) or \(m \geq n(\lfloor \frac{n}{6} \rfloor + \lfloor \frac{n \mod 6}{3} \rfloor)\), then the maximum toughness is half of the maximum connectivity. The same conclusion is obtained if \(r = \lfloor \frac{2m}{n} \rfloor \geq 1\) and \(\frac{(n-1)(r+1)}{2} \leq m < \frac{n(r+1)}{2}\). However, maximum toughness can be strictly less than half of maximum connectivity. Some values of maximum toughness are computed for \(1 \leq n \leq 12\), and some open problems are presented

We describe a random variable \(\text{D}_\text{{n,m}}\), \(\text{n} \geq \text{m} \geq 1\), as the number of failures until the first success in a sequence of n Bernoulli trials containing exactly m successes, for which all possible sequences containing m successes and n-m failures are equally likely. We give the probability density function, the expectation, and the variance of \(\text{D}_\text{{n,m}}\). We define a random variable \(\text{D}_\text{n}\), \(\text{n} \geq 1\), to be the mean of \(\text{D}_\text{n,1}, \ldots, \text{D}_\text{n,n}\). We show that E\([\text{D}_\text{n}]\) is a monotonically increasing function of n and is bounded by \(\ln\) n. We apply these results to a practical application involving a video-on-demand system with interleaved movie files and a delayed start protocol for keeping a balanced workload.

The exact values of \(c(n)\) are determined, where \(c(n)\) denotes the largest \(k\) for which there exists a triangle-free \(k\)-regular graph on \(n\) vertices containing a cut-vertex. As a corollary, we obtain a lower bound on the densest triangle-free regular graphs of given order that do not have a one-factorization.

In the search for doubly resolvable Kirkman triple systems of order \(v\), systems admitting an automorphism of order \((v-3)/3\) fixing three elements, and acting on the remaining elements in three orbits of length \((v-3)/3\), have been of particular interest. We have established by computer that 100 such Kirkman triple systems exist for \(v=21\), 90,598 for \(v=27\), at least 4,494,390 for \(v=33\), and at least 1,626,684 for \(v=39\). This improves substantially on known lower bounds for numbers of Kirkman triple systems. We also establish that the KTS(27)s so produced yield 47 nonisomorphic doubly resolved KTS(27)s admitting the same automorphism.