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.

Fischer classified the groups generated by a conjugacy class 𝐷 of 3-transpositions (involutions such that the product of any two has order at most 3) and discovered three new sporadic groups that bear his name. These groups are rank 3 groups: 𝐷 carries in a natural way the structure of a geometry with lines of length 3 and the structure of a rank 3 graph. We review some properties of these geometries, called Fischer spaces, and mention Fischer’s group-theoretic classification. We discuss the examples and give detailed parameter information. We briefly discuss cotriangular spaces and Shult’s classification, and Hall’s classification of copolar spaces. Finally we classify with full proofs all lax embeddings of (finite) symplectic copolar spaces in projective space. We use some specific such embedded copolar spaces to construct two rank 3 graphs of affine type, one with 2401 vertices, the other with 6561 vertices.

In this short chapter we discuss the p-ranks of matrices related to (strongly regular) graphs. The p-rank of an integral matrix is the rank over the finite field of order p. Designs or graphs with the same parameters can sometimes be distinguished by considering the 𝑝-rank of associated matrices. For strongly regular graphs the interesting primes p are those dividing r-s (where r and s are the eigenvalues distinct from the valence), otherwise the p-rank is completely determined by the parameters. We list the interesting p-ranks of many graphs and discuss also some families of graphs, such as triangular graphs, Paley and Peisert graphs, symplectic graphs. We also discuss the Smith normal form of the adjacency matrix of some families of graphs, such as the complete graphs, lattice graphs, triangular graphs, Paley and Peisert graphs.

The classification of rank 3 graphs is due to Foulser-Kallaher-Kantor-Liebler-Liebeck- Saxl and others. We describe that result in this chapter. We provide the relevant group-theoretic theorems (without proof) and identify all graphs that appear with graphs that were mentioned (and in most cases explicitly constructed) in previous chapters. This way, we give all pairs (Γ,), with Γ a strongly regular graph and 𝐺 a group of automorphisms of Γ acting rank 3. We also provide a table of the parameters of the sporadic rank 3 graphs, which have up to 531441 vertices, and a similar table for all rank 3 graphs up to 1024 vertices. Finally, we tabulate small primitive rank 4–10 strongly regular graphs which do not have Latin square parameters.

In this chapter we discuss strongly regular graphs arising from non-degenerate reflexive forms on a finite vector space not isomorphic to the point-graphs. These comprise the graphs on non-singular or nonisotropic points with respect to reflexive forms over small fields, the half spin graphs with a discussion of triality, the rank 5 hyperbolic case, the graph on the maximal subspaces of rank 3 orthogonal and symplectic polar spaces, affine polar graphs including a detailed discussion of the affine half spin graphs, forms graphs, affine graphs with a Baer subspace or a hyperoval at infinity, and Grassmann graphs. In the last paragraph, we analyse the polar graphs arising from non-isotropic points with respect to various forms over the field of two elements and discuss the mutual relations via the local structure.

In this chapter we construct the Golay codes and the Witt designs, both in several ways. The uniqueness is proved in a self-contained way for the binary case; in the ternary case some details are left out. We then study the associated Witt designs, which are remarkable Steiner systems on 12 and 24 points. We show uniqueness of these, and of the (multiply) derived designs. We define the two standard near polygonsfrom the ternary Golay code and the large Witt design. We discuss the geometry of the projective plane of order 4 providing an alternative construction and uniqueness proof of the Witt designs. Finally, we introduce the Leech lattice and its binary and complex variants.

The first chapter contains the basics of the theory of strongly regular graphs. In particular all basic notions such as parameters and spectrum are rigorously defined. The standard example such as Johnson graphs, Hamming graphs, Paley graphs are introduced. We treat Seidel switching and regular two-graphs, (induced) subgraphs, strongly regular graphs with smallest eigenvalue —2, regular partitions and regular (intriguing) sets. We enumerate the small examples and discuss prolific constructions. This chapter also contains an introduction to two slightly more general objects needed in the book: distance regular graphs (including the main examples, and a discussion on imprimitivity), and association schemes and coherent configurations (including a brief discussion of the Bose-Mesner algebra, linear programming bound, code-clique theorem, Krein parameters, Euclidean representation, subschemas, the absolute bound and the mu-bound).

The second chapter contains some theory of polar spaces. The latter provide the main examples of rank 3 graphs, and, more generally, many examples of strongly regular graphs. Polar spaces are introduced in general as embedded in a projective space, and a full proof of the classification of finite embedded polar spaces is provided.In this chapter, the emphasis is on the graphs whose vertices are objects of a polar space such as the point graph and graphs on (one type of) (maximal) singular subspaces. We review the three types of polar spaces, orthogonal, unitary and symplectic using reflexive forms and discuss Witt’s theorem. For the collinearity graphs of (finite) polar spaces, we discuss, sometimes in detail, sometimes merely surveying, maximal cliques, maximal co-cliques (ovoids), intriguing sets (tight sets and h-ovoids), spreads, partial spreads, hemisystems,…

In this chapter, we describe the sporadic rank 3 graphs, and further interesting strongly regular graphs that have special properties not shared by the other graphs in the infinite families to which they belong. Part of the information given here was obtained using the computer algebra system GAP and its package GRAPE(with Nauty). Among the listed properties one finds (various) constructions, a discussion of the maximal cliques and cocliques, a table with regular sets arising from a group action,local characterizations, sporadic properties such as connections with Ramsey numbers, chromatic number, p-ranks, decompositions, etc. The chapter has exactly 100 sections, mostly each dealing with an individual graph, but often also discussing a small family of graphs with equally many vertices (ranging from 5 to 531441 vertices). It contains some new geometric and/or combinatorial constructions of a few sporadic (exceptional and extraspecial) rank 3 graphs.