We consider certain resolvable designs which have applications to doubly perfect Cartesian authentication schemes. These generalize structures determined by sets of mutually orthogonal Latin squares and are related to semi-Latin squares and other designs which find applications in the design of experiments.

A $1$-spread of a BIBD $\mathcal{D}$ is a set of lines of maximal size of $\mathcal{D}$ which partitions the point set of $\mathcal{D}$. The existence of infinitely many non-symmetric BIBDs which (i) possess a $1$-spread, and (ii) are not merely a multiple of a symmetric BIBD,
is shown. It is also shown that a $1$-spread $\mathcal{S}$ gives rise to a regular group divisible design $\mathcal{G}(\mathcal{S})$. Necessary and sufficient conditions that the dual of such a group divisible design $\mathcal{G}(\mathcal{S})$ be a group divisible design are established and used to show the existence of an infinite class of symmetric regular group divisible designs whose duals are not group divisible.

We consider the changing and unchanging of the edge covering and edge independence numbers of a graph when the graph is modified by deleting a node, deleting an edge, or adding an edge. In this paper, we present characterizations for the graphs in each of these classes and some relationships among them.

Let $G$ be the automorphism group of an $(3,5,26)$ design. We show the following: (i) If $13$ divides $|G|$, then $G$ is a subgroup of $Z_2\times F_{r_{13\cdot 12}}$, where $F_{r_{13\cdot 12}}$ is the Frobenius group of order $13\cdot 12$; (ii)If $5$ divides $|G|$, then $G\cong Z_5$ or $G\cong D_{10}$; and (iii) Otherwise, either $|G|$ divides $3\cdot 2^3$ or $2^4$.

We investigate the edge-gracefulness of $2$-regular graphs.

For $n$ a positive integer and $v$ a vertex of a graph $G$, the $n$th order degree of $v$ in $G$, denoted by ${\text{deg}}_n(v)$, is the number of vertices at distance $n$ from $v$. The graph $G$ is said to be $n$th order regular of degree $k$ if, for every vertex $v$ of $G$, ${\text{deg}}_n(v)=k$. For $n\in \{7,8,\dots ,11\}$, a characterization of $n$th order regular trees of degree $2$ is obtained. It is shown that, for $n\ge 2$ and $k\in \{3,4,5\}$, if $G$ is an $n$th order regular tree of degree $k$, then $G$ has diameter $2n–1$.

We prove that there exist precisely $459$ pairwise non-isomorphic Steiner systems $S(5,6,48)$ stabilized by the group ${PSL}_2(47)$.

The known generalized quadrangles with parameters $(s,t)$ where $|s-t|=2$ have been characterized in several ways by M. De Soete $[D]$, M. De Soete and J. A. Thas $[DT1]$, $[DT2]$, $[DT4]$, and the present author $[P]$. Certain of these results are interpreted for a coset geometry construction.

In this paper, we illustrate the relationship between profiles of Hadamard matrices and weight distributions of codes, give a new and efficient method to determine the minimum weight $d$ of doubly even self-dual $[2n,n,d]$ codes constructed by using Hadamard matrices of order $n=8t+4$ with $t\ge 1$, and present a new proof that the $[2n,n,d]$ codes have $d\ge 8$ for all types of Hadamard matrices of order $n=8t+4$ with $t\ge 1$. Finally, we discuss doubly even self-dual $[72,36,d]$ codes with $d=8$ or $d=12$ constructed by using all currently known Hadamard matrices of order $n=36$.

We define an $extremalgraph$ on $v$ vertices to be a graph that has the maximum number of edges on $v$ vertices, and that contains neither $3$-cycles nor $4$-cycles.
We establish that every vertex of degree at least $3$, in an extremal graph of at least $7$ vertices, is in a $5$-cycle; we enumerate all of the extremal graphs on $21$ or fewer vertices; and we determine the size of extremal graphs of orders $25$, $26$, and $27$.