On the Extremal Regular Directed Graphs without Commutative Diagrams and Their Applications in Coding Theory and Cryptography

We use term regular directed graph (r. d. g.) for the graph of irreflexive binary relation with the constant number outputs (or inputs) for each vertex. The paper is devoted to studies of maximal size ER(d, v) of r. d. g. of order v without commutative diagrams formed by two directed passes of length < d with the common starting and ending points. We introduce the upper bound for ER(d, v), which is one of the analogs of well known Even Circuit Theorem by P. Erdös’. The Erdös’ theorem establish the upper bound on maximal size of simple graphs without cycles of length 2n. It is known to be sharp for the cases n = 2, 3and5 only. The situation with the upper bound for Ed(v) is different: we prove that it is sharp for each d ≥ 2. We introduce the girth of directed graph and establish tight upper and lower bounds on the order of directed cages, i.e. directed regular graphs of given girth and minimal order. The studies of regular directed graphs of large size (or small order) without small commutative diagrams, especially algebraic explicit constructions of them, are motivated by their applications to the design of turbo codes in Coding Theory and cryptographical algorithms. We introduce several new algebraic constructions of directed extremal graphs based on biregular generalized polygons, family of directed graphs of large girth with fixed degree.