An (r,z,k)-mixed graph G has every vertex with undirected degree r, directed in- and out-degree z, and diameter k. In this paper, we study the case r = z = 1, proposing some new constructions of (1,1,k)-mixed graphs with a large number of vertices N. Our study is based on computer techniques for small values of k and the use of graphs on alphabets for general k. In the former case, the constructions are either Cayley or lift graphs. In the latter case, some infinite families of (1,1,k)-mixed graphs are proposed with diameter of the order of 2log2 N.

A graph G of order nv where n≥2 and v≥2 is said to be weakly (n,v)-clique-partitioned if its vertex set can be decomposed in a unique way into n vertex-disjoint v-cliques. It is strongly (n,v)-clique-partitioned if in addition, the only v-cliques of G are the n cliques in the decomposition. We determine the structure of such graphs which have the largest possible number of edges.