In this chapter we give a table with the feasible parameter sets of arbitrary strongly regular graphs on at most 512 vertices, and add comments about the known examples. These include: existence (and number of nonisomorphic examples), the parameters, the spectrum, information about being a descendant of a regular two-graph or being in the switching class of a regular two-graph, possible relation with a projective two-weight code,possible relation with a partial geometry, whether it is a conference graph or transversal design. Miscellaneous comments include references to earlier parts of the book, name of the graph, reason for non-existence, possible relation with a Steiner system, etc.

In the chapter we introduce (spherical) buildings. We develop the theory in some detail, sometimes providing proofs. We introduce the shadow geometries and discuss some properties of particular instances in detail. To that end we use “chain calculus”, which provides an efficient way to determine the diameter of a given shadow geometry, or the maximal distance between two generic objects of distinct type. We hence deduce that the shadow geometry of type E(6,1) yields a strongly regular graph. We provide an explicit construction of that geometry using a split octonion algebra. We also discuss the Klein correspondence, and we discuss triality, again with the aid of a split octonion algebra, and use this to construct the split Cayley hexagon over any field.We deduce a rank 4 representation of a corresponding strongly regular graph.

In this chapter, we look at graphs defined by a difference set in a usually abelian group. Difference sets in a vector space that are invariant under multiplication by scalars are equivalent to two-weight codes and to two-character subsets of a projective space. We survey a lot of examples of such two-character sets (infinite families and sporadic ones, the latter summarised in a table). We review cyclic codes, in particular cyclic two-weight codes and introduce the related Van Lint-Schrijver graphs, the Hill graph, the De Lange graphs and the Peisert graphs. Then our attention goes to the one-dimensional affine rank 3 graphs, which we review in some detail, including proofs of the parameter restrictions that lead to the different cases: the Paley graphs, the Van Lint-Schrijver graphs and the Peisert graphs. We also discuss the Paley graphs in some detail and provide a table with small strongly regular power residue graphs. The penultimate section is dedicated to graphs related to the action of the alternating group Alt(5) and the symmetric group Sym(4) on a projective line. In the last section, we review strongly regular graphs constructed from bent functions.

This chapter collects constructions of strongly regular graphs related to some combinatorial setting, where the starting point is not a group. It discusses Hadamard and conference matrices, (mutually orthogonal) Latin squares, symmetric designs, transversal 3-designs, quasi-symmetric designs (including a table of exceptional parameter sets for such designs with up to 100 points, and we review and prove some results ruling out certain parameter sets of those), partial geometries (including a full proof of Bruck’s and Bose’s sufficient conditions for a graph to be the point graph of a partial geometry, and of Neumaier’s ‘claw bound’), semi-partial geometries and partial quadrangles, (regular) two-graphs, pseudo-cyclic association schemes, and spherical designs. We also briefly discuss the t-vertex condition, asymptotic and randomness properties, the chromatic number and index,and directed strongly regular graphs.