On the homogeneous algebraic graphs of large girth and their applications

Families of finite graphs of large girth were introduced in classical extremal graph theory. One important theoretical result here is the upper bound on the maximal size of the graph with girth ⩾2d established in Even Circuit Theorem by P. Erdös. We consider some results on such algebraic graphs over any field. The upper bound on the dimension of variety of edges for algebraic graphs of girth ⩾2d is established. Getting the lower bound, we use the family of bipartite graphs D(n,K) with n⩾2 over a field K, whose partition sets are two copies of the vector space Kn. We consider the problem of constructing homogeneous algebraic graphs with a prescribed girth and formulate some problems motivated by classical extremal graph theory. Finally, we present a very short survey on applications of finite homogeneous algebraic graphs to coding theory and cryptography.