Seminar/Colloquia Event Item 5306

It is well known that the number of distinct eigenvalues of the adjacency matrix of a graph is a strict upper bound for the diameter of the graph. This bound fails for digraphs, which we demonstrate by constructing digraphs with just three distinct adjacency eigenvalues that have arbitrarily large diameter. We then utilize a vertex-splitting operation that fixes the nonzero spectrum of a digraph while forcing its diameter to grow without bound, and we exhibit explicit families of irregular and regular digraphs in which the gap between the diameter and the number of distinct eigenvalues can be made arbitrarily large.