ABSTRACT: One of the founding problems of algebraic graph theory is that of finding the maximum number of equiangular lines in Euclidean d-space, which is the maximum
size of a simplex in real projective space. In the late 1960s, Van Lint and Seidel expressed this geometric problem in graph-theoretic terms: The Seidel Matrix S(X) of
a graph X is defined by  S(X) = I + 2A(X) - J, in which I is the identity matrix, J is the all-ones matrix, and A(X) is the adjacency matrix of X. Van Lint and Seidel
showed that the problem of finding a set of n equiangular lines in d-space is equivalent to the problem of finding graphs on n vertices whose least Seidel matrix
eigenvalue has multiplicity n-d. They derived a bound on the size of a set of equiangular lines in d-space with mutual angle $\theta$, called the "relative bound"
(relative to $\theta$), and showed that sets of equiangular lines which meet this bound correspond to distance regular double covers of the complete graph, called regular
 two-graphs. Consequently, in the late 60s and early 70s, many algebraic graph theorists found constructions for regular two-graphs with the aim of constructing
large sets of equiangular lines. 

In this talk we will take the opposite approach. We will use linear algebraic methods to construct sets of equiangular lines which meet Seidel's relative bound,
and use them to construct regular two-graphs with cliques of specified order. We show that the existence of a regular two-graph with least eigenvalue $\tau$
containing a clique of order d depends on the existence of an incidence structure on d points with special properties. Quasi-symmetric designs provide examples of these
incidence structures, and we will use them to construct several regular two-graphs. Regular two-graphs have many interesting combinatorial properties. They are the
first nontrivial class of antipodal distance regular graphs, which have applications to other areas of combinatorics, including finite geometry and coding theory. 

This talk is based on my master's thesis research, supervised by Dr. Chris Godsil at the University of Waterloo, supported by NSERC.