We enumerate by computer algorithms all simple \(t-(t+7, t+1, 2)\) designs for \(1 \leq t \leq 5\), i.e., for all possible \(t\). This enumeration is new for \(t \geq 3\). The number of nonisomorphic designs is equal to \(3, 13, 27, 1\) and \(1\) for \(t = 1, 2, 3, 4\) and \(5\), respectively. We also present some properties of these designs, including orders of their full automorphism groups and resolvability.

Let \(G\) be a finite simple graph. The vertex clique covering number \({vcc}(G)\) of \(G\) is the smallest number of cliques (complete subgraphs) needed to cover the vertex set of \(G\). In this paper, we study the function \({vcc}(G)\) for the case when \(G\) is \(r\)-regular and \((r-2)\)-edge-connected. A sharp upper bound for \({vcc}(G)\) is determined. Further, the set of possible values of \({vcc}(G)\) when \(G\) is a \(4\)-regular connected graph is determined.

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 \(n \in \{7, 8, \ldots, 11\}\), a characterization of \(n\)th order regular trees of degree \(2\) is obtained. It is shown that, for \(n \geq 2\) 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.