DMS Combinatorics Seminar

Speaker: Jared DeLeo (Auburn University)

Title: Edge-Regular Graphs and Uniform Shared Neighborhood Structures

Abstract: We explore various results about edge-regular graphs, which are a superset of the well-studied class of strongly regular graphs, and special induced subgraphs within these edge-regular graphs called ‘uniform shared neighborhood structures' (USNS for short). In particular, there exist graphs $H$ such that $H$ is not the USNS for any edge-regular graph $G$, known as a USNS-forbidden graph. We will discuss these USNS-forbidden graph results, as well as using known graph products, including a “new" graph product, to produce new edge-regular graphs that have USNS. The “new" graph product, the unary shadow graph operator, is given special attention and is examined for regular, edge-regular, and strongly regular graphs.