Speaker
Description
In 1985, Brouwer and Mesner proved that the vertex-connectivity of a strongly regular graph equals its valency and the only disconnecting sets of this size are point neighborhoods. In 2009, Brouwer and Koolen generalized this result to distance-regular graphs. In 1996, Brouwer conjectured that the minimum size of a disconnecting set of vertices whose removal disconnects a connected strongly regular graph into non-singleton components equals the size of the neighborhood of an edge. In 2014, Cioaba, Kim, and Koolen disproved Brouwer's conjecture, but also showed that the conjecture is true for many families of strongly regular graphs. In their 2016 survey, van Dam, Koolen, and Tanaka asked whether Brouwer's conjecture is true for distance-regular graph with diameter at least three. In this talk, I will describe our progress regarding this restricted vertex-connectivity of distance-regular graphs.