SOME FAMILIES OF GRAPHS, HYPERGRAPHS AND DIGRAPHS DEFINED BY SYSTEMS OF EQUATIONS: A SURVEY FELIX LAZEBNIK AND SHUYING SUN Abstract. The families of graphs defined by a certain type of system of equa- tions over commutative rings have been studied and used since 1990s, and the only survey of these studies appeared in 2001. In this paper we mostly con- centrate on the related results obtained in the last fifteen years, including generalizations of these constructions to digraphs and hypergraphs. We also offer a unified elementary (i.e., Lie algebra free) exposition of the properties of a family of graphs known as D(k, q). The original results on these graphs appeared in several papers, with the notations reflecting their origins in Lie algebras. The components of graphs D(k, q) provide the best known general lower bounds for the number of edges in graphs of given order and given girth (the length of a shortest cycle). The paper also contains several open problems and conjectures. 1. Introduction One goal of this survey is to summarize results concerning certain families of graphs, hypergraphs and digraphs defined by certain systems of equations, concen- trating on the results which appeared during the last fifteen years. Another goal is to provide a comprehensive treatment of, probably, the best known family of such graphs, denoted by D(k,q), including most of related (and revised) proofs. The original results on these graphs were scattered among many papers, with the notations not necessarily consistent and reflecting the origins of these graphs in Lie algebras. It is our hope that this new exposition will make it easier for those who wish to understand the methods, continue research in the area or find new applications. For a summary of related results which appeared before 2001, see Lazebnik and Woldar [71]. One important feature of that article was an attempt of setting simpler notation and presenting results in greater generality. The current presentation is based on that paper. Let us begin with a quote from [71] (with updated reference labels): In the last several years some algebraic constructions of graphs have appeared in the literature. Many of these constructions were motivated by problems from extremal graph theory, and, as a con- sequence, the graphs obtained were primarily of interest in the con- text of a particular extremal problem. In the case of the graphs Date : September 14, 2016. Key words and phrases. Girth, embedded spectra, lift of a graph, cover of a graph, edge- decomposition, isomorphism, generalized polygons, digraph, hypergraph, degenerate Tur´an-type problems. 1