**On new results on Extremal Graph Theory, Theory of Algebraic Graphs and their applications in Cryptography and Coding Theory.**

*Vasyl Ustimenko*

**Abstract: **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 commutative ring K.
For each commutative integrity ring K with |K|>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 K^n, n>1 and cycle indicator ≥ 2n+2 such that their projective limit is well defined and isomorphic to an infinite forest.

**Category / Keywords: **foundations / family of graphs of large girth, small world graphs, cryptographic algorithms, LDPC codes.

**Original Publication**** (with minor differences): **will be published in Dopovidi of Nath. Acad. of Ukraine

**Date: **received 3 Mar 2022

**Contact author: **vasyl at hektor umcs lublin pl

**Available format(s): **PDF | BibTeX Citation

**Version: **20220307:124224 (All versions of this report)

**Short URL: **ia.cr/2022/296

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