Title: Algebraic Graph Theory
|Affiliation:||University of Waterloo|
A graph is distance-regular if we can write the distance adjacency matrices as polynomials in the adjacency matrix. Distance-regular graphs are a class of graphs of significant interest to algebraic graph theorists for their structural and algebraic properties. The notion of distance-regularity can be weakened to a local property on vertices, but when every vertex in the graph is locally distance-regular, the graph will either be distance-regular or in the closely related class of distance-biregular graphs. In fact, we can weaken the notion of locally distance-regular further to pseudo-distance-regular vertices while still preserving the property that a graph where every vertex is pseudo-distance-regular must be either distance-regular or distance-biregular. Then by finding characterizations of pseudo-distance-regular vertices, we can scale this up to characterizations of distance-regular graphs. In this talk, we will present several such characterizations, based on work by Fiol, Garriga, and Yebra.