@article{Ustimenko_2022, place={Kyiv, Ukraine}, title={On new results on extremal graph theory, theory of algebraic graphs, and their applications}, url={https://dopovidi-nanu.org.ua/ojs/index.php/dp/article/view/2022-4-4}, DOI={10.15407/dopovidi2022.04.025}, abstractNote={<p>New explicit constructions of infinite families of finite small world graphs of large girth with well-defined projective limits which is an infinite tree are described. The applications of these objects to constructions of LDPC codes and cryptographic algorithms are shortly observed. We define families of homogeneous algebraic graphs of large girth over the commutative ring K. For each commutative integrity ring K with |K| &gt; 2, we introduce a family of bipartite homogeneous algebraic graphs of large girth over K formed by graphs with sets of points and lines isomorphic to K<sup>n</sup>, n &gt; 1, and cycle indicator ≥ 2n + 2 such that their projective limit is well defined and isomorphic to an infinite forest.</p>}, number={4}, journal={Reports of the National Academy of Sciences of Ukraine}, author={Ustimenko, V.O.}, year={2022}, month={Aug.}, pages={25–32} }