Algebraic Graph Theory Seminar - Soffia Arnadottir

Title: Strongly cospectral vertices in cubelike graphs

Abstract:

A cubelike graph is a Cayley graph of the elementary abelian 2-group. Two vertices in a graph are strongly copsectral if they are cospectral and parallel. It is easy to find cubelike graphs with strongly cospectral vertices; any cubelike graph for which the sum of the connection set is non-zero has perfect state transfer, and thus it has pairs of strongly cospectral vertices. But can we find cubelike graphs with larger sets of vertices that are pairwise strongly cospectral? (Spoiler: yes). I will show that for all $d\geq5$, there are cubelike graphs on $2^d$ vertices with sets of four pairwise strongly cospectral vertices.