The ACO Seminar (2021–2022)

A covering graph is a structure obtained from a graph by replacing every vertex with a coclique of size $r$ and replacing every edge with a perfect matching. It can be shown that a cover of $X$ has the same spectrum as $X$ plus (possibly) some new eigenvalues. It turns out that when a cover has exactly two new eigenvalues (2-ev cover), we can show that this cover inherits some structural properties from the original graph. In this talk, I will prove that a 2-ev cover of a walk-regular graph is walk-regular and that a 2-ev cover of a complete graph is distance-regular. This is a joint work with Chris Godsil and Max Levit from the University of Waterloo.