A generalization of Sperner’s labeling for simplices is considered. It allows us to give any label not only to points from the interior of the simplex but also to points from the relative interior of each facet, while the Sperner labeling rule is preserved for all points on the boundary of each facet. Some properties of this labeling and its behavior on the facets of the simplex are discussed. Also, necessary and sufficient conditions for the existence of an odd number of completely labelled simplices in any triangulation of the simplex are given.

Orthomorphism graphs of groups are defined and a correspondence, between cliques of orthomorphism graphs and difference matrices and generalized Hadamard matrices, is established. Some examples of orthomorphism graphs are given.
Also, for \(\lambda = 1\), known values and bounds for clique numbers of orthomorphism graphs of groups of small order are surveyed.

In this paper we consider the problem of characterizing directed graphs of specified diameter. We are especially interested in the minimal number of arcs \(\textbf{a(d,n)}\) required to construct a directed graph on \(n\) vertices with diameter \(d\). Classes of graphs considered include general digraphs, digraphs without cycles of length \(2\), and digraphs with regular indegree or regular outdegree. Upper bounds are developed in cases where the exact solutions are not known.

In assessing the “vulnerability” of a graph one determines the extent to which the graph retains certain properties after the removal of a number of vertices and/or edges. Four measures of vulnerability to vertex removal are compared for classes of graphs with edge densities ranging from that of trees to that of the complete graph.

Lander conjectured: If D is a \((\text{v,k},\lambda)\) difference set in an abelian group \(G\) with a cyclic Sylow \(p\)-subgroup, then \(p\) does not divide \((v,n)\), where \(\text{n = k}-\lambda\).

Various nonexistence theorems are used to verify the above conjecture (all hand calculations) for \(\text{k} \leq 500\), except for \(\text{k} = 228, 282\) and \(444\), when \(\lambda = 3\). Using a machine, it is possible to do the checking for large \(k\).