This paper describes some new methods of constructing rectangular designs from balanced incomplete block (BIB) designs and Hadamard matrices. At the end of the paper, a table of rectangular designs in the range of \(r\),\(k \leq 15\) is given.

An \(n \times n\) sign pattern \(A\) is a spectrally arbitrary pattern if for any given real monic polynomial \(f(x)\) of degree \(n\), there is a real matrix \(B \in Q(A)\) having characteristic polynomial \(f(x)\). In this paper, we give two new classes of \(n \times n\) spectrally arbitrary sign patterns which are generalizations of the pattern \(W_{n}(k)\) defined in [T. Britz, J.J. McDonald, D.D. Olesky, P. van den Driessche, Minimal spectrally arbitrary sign patterns, SIAM Journal on Matrix Analysis and Applications, \(26(2004), 257-271]\).

We show that the power subgroups \(M^{6k}\) (\(k > 1\)) of the Modular group \(M = \text{PSL}(2, \mathbb{Z})\) are subgroups of the groups \(M'(6k, 6k)\). Here, the groups \(M'(6k, 6k)\) (\(k > 1\)) are subgroups of the commutator subgroup \(M’\) of \(M\) of index \(36k^2\) in \(M’\).

Let \(G\) be a simple graph. The double vertex graph \(U_2(G)\) of \(G\) is the graph whose vertex set consists of all \(2\)-subsets of \(V(G)\) such that two distinct vertices \(\{x,y\}\) and \(\{u,v\}\) are adjacent if and only if \(|\{x,y\} \cap \{u,v\}| = 1\) and if \(x = u\), then \(y\) and \(v\) are adjacent in \(G\). In this paper, we consider the exponents and primitivity relationships between a simple graph and its double vertex graph. A sharp upper bound on exponents of double vertex graphs of primitive simple graphs and the characterization of extremal graphs are obtained.

Let \(S_n\) be the set of permutations on \(\{1, \ldots, n\}\) and \(\pi \in S_n\). Let \(d(\pi)\) be the arithmetic average of \(\{|i – \pi(i)| : 1 \leq i \leq n\}\). Then \(d(\pi)/n \in [0, 1/2]\), the expected value of \(d(\pi)/n\) approaches \(1/3\) as \(n\) approaches infinity, and \(d(\pi)/n\) is close to \(1/3\) for most permutations. We describe all permutations \(\pi\) with maximal \(d(\pi)\).

Let \(s^+(\pi)\) and \(s^*(\pi)\) be the arithmetic and geometric averages of \(\{|\pi(i) – \pi(i + 1)| : 1 \leq i 1\). We describe all permutations \(\pi\),\(\sigma\) with maximal \(s^+(\pi)\) and \(s^*(\sigma)\).

A connected graph \(G = (V,E)\) is said to be \((a,d)\)-antimagic, for some positive integers \(a\) and \(d\), if its edges admit a labeling by all the integers in the set \(\{1, 2, \ldots, |E(G)|\}\) such that the induced vertex labels, obtained by adding all the labels of the edges adjacent to each vertex, consist of an arithmetic progression with the first term \(a\) and the common difference \(d\). Mirka Miller and Martin Bača proved that the generalized Petersen graph \(P(n,2)\) is \((\frac{3n+6}{2}, 3)\)-antimagic for \(n \equiv 0 \pmod{4}\), \(n \geq 8\), and conjectured that \(P(n, k)\) is \((\frac{5n+5}{2}, 2)\)-antimagic for odd \(n\) and \(2 \leq k \leq \frac{n}{2}-1\). In this paper, we show that the generalized Petersen graph \(P(n,2)\) is \((\frac{5n+5}{2}, 2)\)-antimagic for \(n \equiv 3 \pmod{4}\), \(n \geq 7\).

Sierpiński graphs \(S(n,k)\), \(n, k \in \mathbb{N}\), can be interpreted as graphs of a variant of the Tower of Hanoi with \(k \geq 3\) pegs and \(n \geq 1\) discs. In particular, it has been proved that for \(k = 3\) the graphs \(S(n, 3)\) are isomorphic to the Hanoi graphs \(H_3^n\). In this paper, we will determine the chromatic number, the diameter, the eccentricity of a vertex, the radius, and the centre of \(S(n,k)\). Moreover, we will derive an important invariant and a number-theoretical characterization of \(S(n,k)\). By means of these results, we will determine the complexity of Problem \(1\), that is, the complexity of getting from an arbitrary vertex \(v \in S(n,k)\) to the nearest and to the most distant extreme vertex. For the Hanoi graphs \(H_3^n\), some of these results are new.

In this paper, we will prove that there exist no \([n,k,d]_q\) codes of \(sq^{k-1}-(s+t)q^{k-2}-q^{k-4} \leq d \leq sq^{k-1}-(s+t)q^{k-2}\) attaining the Griesmer bound with \(k \geq 4, 1 \leq s \leq k-2, t \geq 1\), and \(s+t \leq (q+1)\backslash 2\). Furthermore, we will prove that there exist no \([n,k,d]_q\) codes for \(sq^{k-1}-(s+t)q^{k-2}-q^{k-3} \leq d \leq s\) attaining the Griesmer bound with \(k \geq 3\), \(1 \leq s \leq k-2\), \(t \geq 1\), and \(s+t \leq \sqrt{q}-1\). The results generalize the nonexistence theorems of Tatsuya Maruta (see \([7]\)) and Andreas Klein (see \([4]\)) to a larger class of codes.