Combinatorial methods for constructing bipartite uniform minimal quasicomplete graphs (symmetrical block designs)

Consideration was given to a specific family of bipartite graphs consisting of two disjoint subsets $X$ and $Y$ of vertices and characterized by that each vertex in $X(Y)$ is connected to each of the remaining vertices in $X(Y)$ by a unique path of length two passing through some vertex in $Y(X)$. The prefix “quasi” reflects the fact that complete connection of the vertices is realized by paths of length two rather than by edges. The problem of constructing uniform minimal graphs with identical cardinalities of the subsets $X$ and $Y$ which is of practical interest for complex communication networks was discussed. It belongs to the class of combinatorial problems of construction of the so-called symmetrical block designs.